Despite ongoing disputes about the subgrouping of the Uralic family, it is clearly the case that the Finnic and Samic languages have been at least neighbors for several millennia now, exchanging linguistic features and material back and forth. With care, this allows teasing out substantial facts about the relative chronology of the history of the two families. (Germanic can be also added to the bundle, though the evidence from here is much more unidirectional.)
The sibilant system shows several good examples. While Finnic /s/ and Samic /s/ correspond to each other consistently all the way from Proto-Uralic to the present day, the “shibilants” have a more complex history. In old inherited vocabulary the main correspondences for these are Finnic *s ~ Samic *č (from original *ś ~ *ć, be it Proto-Finno-Samic or all the way from Proto-Uralic) and Finnic *h ~ Samic *s (from original *š). The latter correspondence can also appear in old (perhaps mostly either parallel or Finnic-mediated) loans from Germanic, whose *s was substituted as *š at least on the Finnic side; no way to tell if also in Samic.
(This probably indicates that pre-Finnic *s was, following the merger of *s and *ś, realized as laminal [s̻], while *š was (sub?)apical [ʂ]. Germanic *s was likely apical [s̺], and therefore matched better with pre-Finnic *š. I am not sure how far back the modern Finnish realization of /s/ as apical [s̺] dates, but at least the Northern Karelian shift of *s to an apical postalveolar [s̱] š most likely starts from this same value.)
The correspondence Finnic *s ~ Samic *š appears in a small number of native-looking cases, where they seem to represent original preconsonantal *ś (PF *laskë- ~ PS *lōštē- < *laśkə- ~ *laśk-ta- ‘to let out, pour, etc.’; PF *kisko- ~ PS *këškē- < *kiśka(w)- ‘to tear, pull’; PF *vaski~ PS *veaškē < *wäśkä ~ *waśka ‘copper’). It is however more common in loanwords between the two. E.g. Finnic *s before *i and *ü seems to be fairly regularly substituted as *š in Samic; the YSS data has 5 examples of this out of its 11 examples altogether of PS *š-.  All late loanwords from Samic into Finnish also show *š → /s/, for the obvious reason that Finnish has had no other sibilants for most of its independent existence. (Even the modern loanword phoneme š or sh is still limited to educated speakers. Probably a rather large proportion of Finns counts as “educated” by typical contact linguistic standards by now, though…)
Lastly, also the fourth theoretically possible correspondence between plain sibilants is attested: Finnic *h < *š ~ Samic *š. (I will not be treating the various affricates in this post.) This might be the group that has the most value for establishing chronology, since it is bounded both from above and below: prevocalic *š only occurs in loanwords in Proto-Samic, but any such loanwords from Finnic must then pre-date the pan-Finnic change *š > *h.
Some of the data in this group suggests that it stretches beyond the breakup of Proto-Samic. One example is the word for ‘coal, ember’; in Finnic *šiili > *hiili (Fi. hiili etc.), which then appears as pseudo-PS *šilë in Southern, Ume and Pite Sami (SS sjïjle etc.); as pseudo-PS *hilë in Lule and Northern Sami (NS hilla); and pseudo-PS *ilë in Eastern Sami (Inari illâ etc.). I’ve sometimes seen also the explanation that these kind of cases would not be parallel loanwords, but rather several layers of re-loaning, with each new loanword then flushing out the previous one. This however seems unlikely to me, especially when dealing with a non-cultural term like ‘coal’ that has no reason to be repeatedly loaned from Finnic, and when the distribution of the different variants is perfectly complementary. 
Meanwhile, *š > *h is usually taken to be late Proto-Finnic, i.e. at least Proto-Core Finnic (probably later than at least the splitting of South Estonian though). Does this mean that Proto-Samic is therefore younger than even Core Finnic? And how does this measure up with how e.g. Jaakko Häkkinen (Jatkuvuusperustelut ja saamelaisen kielen leviäminen, osa 2: see table on p. 19) comes out with the opposite result: Proto-Samic would have broken up earlier than Proto-Finnic?
One option would be to sigh and concede that apparently words like ‘coal’ are multiple layers after all. But I would hold out for a different explanation: we can probably shift the dating of *š > *h ahead quite a bit beyond its various termini post quem. E.g. the introduction of *h → *h in Germanic loanwords into Finnic does not have to be enabled by the development of a native /h/ in Finnic; it can represent also the taking-up of a new loanword phoneme, which besides probably already existed as an allophone in the clusters *kt [ht] and/or *sl *sr *sn [hl hr hn]. In fact, since Proto-Finnic also had all four of *st *kl *kr *kn, then the introduction of [h] in both *kt and the *sR group would have already been sufficient to phonemicize it: it could be no longer identified uniquely as either /s/ or as /k/. — Again, I plan on writing a full article on this topic in the future.
This finally brings me to the topic I mention in the title. The Samic languages have borrowed *š from early Finnic also in several consonant-stem nominals. However, while these have consistently /-š/ in Western Sami, they seem to have dual representation in Eastern Sami: sometimes they surface with /-s/, sometimes with /-š/. At first sight this sounds like it might be related to the fact that some of these cases are loaned from PF *-is and not *-eš — but no, that contrast appears to be completely orthogonal.
Let’s roll out the data:
(1) Eastern Sami /-š/ ← Finnic *-eš
- F *imeš (> Fi. ihme ‘wonder’) → S *imëš > e.g. North imaš, Inari iimâš
- F *kadëš (> Fi. kade ‘jealous’) → S *kāðëš > e.g. North gáđaš, Inari kaađâš
- F *laudëš (> Fi. laude ‘seat in sauna’) → S *lāvtëš > Skolt laaudâš
- F *murëš (> Fi. murhe ‘sorrow’) → S *morëš > e.g. North moraš, Inari muurâš
- F *säigeš (> Fi. säie ‘thread, fiber’) → S *šeajkëš > Kildin šieigaš
- F *säigeš also? → S *sājkëš > e.g. Skolt saaiǥâš
(This looks like a contamination of the previous word × the verb *sājkē- ‘to wear out’ reflected in most of Samic; which is probably not loan, but older inheritance from original *säjkä, as no vowel-stem forms survive in Finnic. North sáiggas ‘worn’ is then simply a native derivative from the verb, as also per the semantics.)
- F *tarbëš (> Fi. tarve ‘need’) → S *tārpëš > e.g. North dárbbaš, Inari taarbâš
(In this one case, with an *s-stem quite widely alongside: *tārpēs > Southern daerbies, also Lule; *tārpës > e.g. Skolt taarbâs, also Pite, Lule. Lule Sami seems to have all three variants: dárpaj, dárpes, dárpas, and even a vowel-stem dárpa. There is Finnish dialectal tarvis as well, so the diversity clearly goes back to parallel loaning in some fashion.)
(2) Eastern Sami /-š/ ← Finnic *-is
- F *kallis (> Fi. kallis ‘expensive’) → S. *kāllëš > e.g. North dial. gállaš, Skolt kaallâš
(in Inari *ēs-stem kaalles, apparently with a nativized adjective ending)
- F *ruumis (> Fi. ruumis ‘corpse’) → S. *rumëš > e.g. North rumaš, Inari ruumâš
(parallel *romës in Skolt roomâs,  compareable with the Fi. dialectal variant rumis from Southern Ostrobothnia; and with a vowel stem in Southern Sami: *romē > räbmie.)
- F *rugis (> Fi. ruis ‘rye’) → S. *rukëš > e.g. North rugaš, Inari ruuvaš
- F *valmis (> Fi. valmis ‘ready’) → S. *vālmëš > e.g. North válmmaš, Inari vaalmâš
(3) Eastern Sami /-s/ ← Finnic *-eš
- F *kantëlëš (> Fi. kantele ‘a traditional string instrument’) → S *kāntëlës > Inari kaddâlâs
- F *kiireš (> Fi. kiire ‘hurry’) → S *kirës > e.g. Inari kiirâs
- F *kärmeš (> Fi. käärme ‘snake’) → S *kearmëš ~ *kearmës > e.g. North gearpmaš, Ter “kermʾs“
- F *pereš (> Fi. perhe ‘family’) → S *pearëš ~ *pierës > e.g. North bearaš, Kildin пӣрас
(vowel-stem *pearë in Skolt piâr)
- F *terveš (> Fi. terve ‘healthy’) → S *tearvëš ~ *tiervëš > e.g. North dearvvaš, Inari tiervâs
- F *voidëš (> Fi. voide ‘lotion, ointment) → S *vōjtës > Inari vuoidâs
(From Sammallahti’s reverse dictionary of Inari Sami. Álgu does not have this lexeme, so I have no idea if there are equivalents elsewhere in Samic. This could be also an independent derivative within Samic from the base verb: PS *vōjtë- ‘to grease, anoint’, interestingly an *ë-stem one instead of *ē-stem, as could be expected.)
(4) Eastern Sami /-s/ ← Finnic *-is
- F *nakris (> Fi. nauris ‘swede (type of turnip)’) → S *nāvrëš ~ *nāvrës > e.g. North návrraš, Inari naavrâs
- F *saalis (> Fi. saalis ‘catch’) → S *sālëš ~ *sālës > e.g. North sálaš, Inari saalâs
A few initial comments:
- I’ve only included cases with -s when Western Sami, or failing that Finnic, actually points to *-š. Of course F *-š ~ S *-s can be also found in older shared vocabulary, as in ‘boat’: *venəš > F. *veneš > Fi. vene; > S. *vënës > e.g. North vanas, Inari voonâs; > Mordvinic *venəš > e.g. Erzya венч /venč/. ‘Hurry’ could be theoretically also of this type; per the vowel correspondence *a ~ *ā, kaddâlâs clearly cannot.
- This entire word group seems to be centered on Northern and Inari Sami. Reflexes are practically absent from Southern Sami (only gïermesj ‘snake’), very rare also in Ume and Pite Sami. This would fit well together with late separate loaning from early Finnish specifically + occasional diffusion into other Sami varieties.
- Some of these words are originally from Germanic, and could be in theory partly borrowed directly from there into Samic, but I haven’t found any examples where Proto(-Western)-Samic *-ëš appears in a loanword without Finnic equivalents. Also, many enough cases are native Finnic, either wholly (e.g. kantele perhe säie, nauris saalis) or at least the *S-derivative is (terve); or come from Baltic (käärme). The only case where parallel loaning is clearly involved is the ‘need’ group: probably *tārpëš via Finnic, versus *tārpës directly from Scandinavian *þarbiz.
Here is a quick distribution chart, as you may wish to consult for point 2: 
*imëš: - - - L N I - - - *kātëš: - - - - N I S - - *lāvtëš: - - - - - - S - - *murëš: - - - L N I S - - *seajkëš: - - - - - - - K - *sājkëš: - - - - - - S K - *tārpëš: - - - L N I - - - *kāllëš: - - - - N I S K T *rumëš: - - - L N I S - - *rukëš: - - - L N I - - - *vālmëš: - - - L N I S K T *kāntëlës: - - - - - I - - - *kirës: - - - - - I S K T *kearmëš/s: S - P L N - - - T *pErëš/s: - - - L N - - K T *tErvëš/s: - - - - N I S K T *nāvrëš/s: - U P L N I S K - *sālëš/s: - - - L N I - - - S U P L N I S K T totals: 1 1 2 10 13 13 10 8 6
OK then, caveats done with, what is actually going on in here?
Mikko Korhonen in Johdatus lapin kielen historiaan mentions passingly (p. 200) only that the *-ëš-group “appears in correspondence to the loan original’s h” (“š esiintyy itämerensuomalaisissa lainoissa originaalin h:ta vastaamassa“). The same is stated in stronger terms by Mikko Heikkilä in Bidrag til fennoskandiens språkliga förhistoriet i tid och rum (p. 107), where he claims that late Proto-Finnic *h would have been adopted in Samic as *h syllable-initially and *š syllable-finally. This seems phonetically implausible to me however, given that (1) Scandinavian /h/ is regularly borrowed into Samic as /h/ ~ ∅, never as **š, (2) Finnic coda *h from *k is never borrowed as Samic **š, and (3) there definitely is also a layer of loanwords where Finnic onset *š gives Samic *š.
Heikkilä seems to suggest that late substitution as *š could have involved loaning from an intermediate stage of the *š >> *h shift, that he gives as [ç]. A palatal fricative could indeed be plausibly borrowed as Proto-Samic *š, especially if this was a palatal sibilant [ɕ] (as suggested by its origin from *ś, and its later development to /j/ in Western Sami, when before a consonant). This intermediate reconstruction is however based on a common misunderstanding. Sound changes of the type *š > *h do not involve a trek through every single intervening POA you can find on an IPA chart!  These are rooted in the tendency of retroflex consonants in particular to acquire a velar coarticulation, which can then take over as the primary POA; and also for spirants such as [x] to lenite to [h]. Palatal [ç] would be overpassed entirely in this process.
So I see no other explanation than that the cases of *-ëš ~ *-eh must have been borrowed before the Finnic sound change *š > *h (before the loss of the sibilant feature, to be exact). And the distribution suggests that Proto-Samic would have been by this point already quite thoroughly broken up: after all, these words seem to have been borrowed independently mainly into the precedessors of L N I S. Perhaps Proto-North-Lule and Proto-Inari-Skolt at the deepest, in case such entities could be assumed (usually classifications of the Sami languages go with Pite-Lule and Skolt-Kola groupings instead, but I am not entirely sold on this).
In other words, I answer my headline question regardless in the negative: no, there did not exist any *š-stems yet in Proto-Samic, not even in any possible early subgroups like Proto-Western, Proto-Eastern or Proto-Non-Southern; they have only come about later through contacts with early Finnic.
I have not invented any real explanation for the dual treatment of *-š in Eastern Sami. For this, I can only offer a few hypotheses (that all point in different directions):
- maybe early on there was a sound change *-š > *-s in Eastern Sami, and cases with retained -š are newer loans, perhaps partly from Northern Sami (since they seem to be fairly rare in Kola Sami)? This probably could not be equated with the general Samic shift of original *š to *s, since there are many enough good examples of the retention of prevocalic PS *š- in Eastern Sami, and none of unexpected *s- (that I know of). 
- maybe Finnic *š was for a while again borrowed in Eastern Sami as *s, due to being increasingly non-palatal [ʂ], while cases with *-š are older loans from an [ʃ] stage?
- maybe a lost Finnic variety has been involved where word-final *-š > *-s? A late analogical development of *-h-stems to -s-stems is known from Southern Ostrobothnia… which is however nowhere near the attested Eastern Sami languages.
Going by the vowel substitutions also diverging in *pearëš ~ *pierës, *tearvëš ~ *tiervës and *rumëš ~ *romës, the last two explanations sound somewhat better than the first.
This problem very likely needs to be further tied in with *-eš ~ *-is variation appearing even within Finnic, again largely with a West-East divide, such as Western Fi. tarve ~ Eastern Fi. tarvis; Fi. käärme ~ Karelian keärmis; Fi. säie ~ Karelian säijis; Fi. laine < *laineh ~ Olonets-Ludian-Veps lainis ‘wave’. But it is not clear to me if this is good enough to run with my third hypothesis, since there seems to be very little correlation in the occurrence of alternation in Eastern Sami versus in Finnic: there is no e.g. **seajkës in Samic, and more importantly, no **kantelis, **peris or **nakrëš, **ruumëš, **saalëš in Finnic.
 YSS has been now added to my fairly slowly growing Bibliography. If anyone’s curious, the five words with *š ← *s(i, ü) are: PS *šëlëtē ← PF *siledä ‘smooth’; Western *šëljō ~ Northern+Eastern *šiljō ← Fi. silja ‘courtyard’ (clearly rather one of the post-PS loanwords); PS *šëlmē ‘eye of an ax’ ← PF *silmä ‘eye’; PS *šëltē ← *silta < PF *cilta ‘bridge’; PS *šëntë- ← PF *süntü- ‘to become, be born’. I also suspect that PS *šōjē ‘rowan’ may derive from PF *sooja ‘protection’, as in Finnish (as also e.g. Germanic) mythology / folk belief the rowan tree has been considered to grant protection to the homestead. It’s not quite clear why would we have *š- and not *s- here, though. An independent loan from the same Indo-Iranian source (*sćāyā- ‘protection’) would also work.
 A slightly better explanation along almost the same lines might be “etymological alienization”, where the existence of Finnish hiili would have prompted a reshaping of e.g. expected Northern Sami ˣšilla into hilla, possibly fairly late then. This does not seem to be feasible in the case of Eastern Sami, though: in particular Inari and Skolt Sami have only come into intensive contact with Finnish fairly late, but the lack of of ˣh- indicates relatively early loaning. (IIUC /h/ → ∅ has remained the default case in contacts between Karelian and Kola Sami, however.)
 Álgu gives for this a comparison with North ruomas ‘wolf’, which looks like a rather recent (taboo? epithet?) borrowing from Skolt.
 S U P L N I S K T for Southern, Ume, Pite, Lule, Northern, Inari, Skolt, Kildin and Ter Sami respectively. Yes, that’s “S” appering twice, but you can figure this out.
 One impressive example of this approach is the development path “ʃ > ʂ > ç > x > χ > ħ> h” given in Kallio’s “Kantasuomen konsonanttihistoriaa”.
 Amusingly but probably unrelatedly: in “An essay on Saami ethnolinguistic prehistory” Ante Aikio mentions five examples of the “opposite” correspondence, with *s- in Western Sami ~ *š- in Eastern Sami.