A long-standing mystery of Uralic historical linguistics is a split representation of Proto-Uralic *ś in Mansi. Aside from confusion between *ś and *ć (widespread across the entire Uralic family), there are also two plain sibilant reflexes: *s and *š.
What I gather is the current best explanation, as stated e.g. in the two handbooks (by Sammallahti in ’88 and Honti in ’98), is *š being the default reflex; *s would have originated by dissimilation before another palatalized consonant, and would then have spread to other positions by some unexplained mechanism. This implies the split would have been initiated during the original depalatalization of *ś, i.e. at the Ugric or possibly even the East Uralic stage.
I think this is completely on the wrong track. On an examination of the evidence, it seems to me that there is a different, fairly simple solution.
To begin, let us consider the word-initial position. Before the PU illabial non-open front vowels *i and *e, leading to Proto-Mansi *i, *ee and *ä, the main development is *ś > *š:
- *śeðäm(ə) > *šim ‘heart’ 
- *śekä > *šiɣ ‘catfish, burbot’ 
- *śenä > *šeenəɣ ‘mushroom’
- *śepä > *šip ‘neck’
- *śilmä > *šäm ‘eye’
- *śilä > *šilt ‘fat (of bear)’ 
Before the PU labial front vowel *ü > PMs *ü, and the open front vowel *ä > PMs *ää, the main development is *ś > *s:
- *śüðʲə > *süĺ ‘coal’ 
- *śülkə- > *süĺɣə- ‘to spit’
- *śäŋ(k)ə- > *sääŋk- ‘to break’
- *śäŋkə > *sääŋkʷ ‘hip, loin’
- *śäxə > *säɣ ‘plait’ 
The development *ś > *s is also frequent before all back vowels:
- *śarta > *surtə ‘(young) reindeer’
- *śarma > *surəm ‘hole in ceiling’
- *śawŋa > *suuw ‘pole’
- *śojə- > *suj- ‘to sound’
- *śajəmɜ > *saajəm ‘brook’
- *śo/ëðka > *sëëĺ ‘duck’
- *śëla- > *sëël- ‘lightning’
- *śëlka > *sëëɣla ‘pole’
- *śëmə > *sëëm ‘fish scales’
- *śura > *sar ‘narrow’
- *śuka > *sow ‘bark’
- *śurə-ma > *sorəm ‘death’
- *śuwə > *su-nt ‘mouth’
- *śuwə > *suw-ĺ ‘clay’
By this point it is easy to craft a hypothesis: at some point in the development of Mansi, PU *ś, or perhaps rather the depalatalized Proto-Ugric *s, was palatalized to *š before the descendants of original *i, *e.  This would have been phonemicized when the lenition *č > *š occurred. I think this already stands well up to the apparently commonly accepted “dissimilation” handwave: I count 6 examples of *s before a palatal/ized consonant, versus 13 before something else, not exactly a strong track record.
Still, some exceptions to the coherent picture above are found as well. I’ve gotten together 12 cases that have proposed cognates outside of Ob-Ugric. Many of them seem likely to not be actually inherited from Proto-Uralic though.
First, in two words it appears that PU *ä yields PMs *i, each with a different development of the initial consonant:
- *śäkśə > *siɣəs ‘gull’
- *śälä- > *šil-t- ‘to cut’
I’ve commented before that the former of these does not have a particularly good claim to being an inherited PU root. The latter word could be assumed to have acquired its non-open vowel sufficiently erly to have participated in the same palatalization as words where *i goes back to PU *e. Though it also seems there is a possibility that /š/ in this word does not go back to Proto-Mansi: the Lower Konda dialect normally retains *š (/šiɣ/ ‘burbot’, /šip/ ‘neck’, etc.), but here it has /silt-/.
One other case of *s being found before *i in Mansi is *sir ‘order, way’. This has cognates in many Uralic languages, most of which incidate PU *i or *e (e.g. Komi /śer/, Mari *sʏr, Hungarian szër), and this appears to be a clear exception. I wonder if loaning from the Khanty cognate *siir could be the explanation however: the vowel correspondence *i ~ *ii is etymologically quite irregular, but would make phonetic sense as a substitution (the vowels in Khanty have a normal : overshort contrast). Moreover, the Mansi word is mostly not attested from the dialects in the least contact with Khanty: Southern Mansi, or any Western dialect other than that of Pelymka.
A truly puzzling word is *säjə or *sejə > *säj ‘pus’. The other Uralic languages are all in quite good agreement that the original initial was plain *s-, not palatalized *ś- (e.g. Samic *sējë, Mordvinic *sij, Khanty *ɬöj, Hungarian ev). No matter how Mansi ended up retaining *s- here, this case cannot tell us anything about the development of PU *ś. 
There is also a bunch of words where *ś > *š is supposed to occur before back vowels. Five of these are more or less widespred, while three others have suggested cognates only in one Uralic branch beside Mansi.
- *śëta > *šëët ‘100’
- *śora- > *šuurl- ‘to dry’
- *śarwə > *šaarəp ‘horn’
- *śosra > *šaatər ‘1000’
- *śuðʲa > *šaĺ ‘frost’
- Samic *čoanē ~ Ms *šun ‘sleigh’
- Finnic *sopa ~ Ms *šup ‘shirt’
- Finnic *sampi ~ Ms *šupəɣ ‘sturgeon’
The last three I think can be discarded offhand. Mansi *u usually derives from PU *u (*luwə > *luw ‘bone’, *purə- > *pur- ‘to bite’, etc.) and does not regularly correspond with Samic *oa or Finnic *o or *a. The last item also has a discrepancy between Finnic *mp and Mansi *p. For the second-to-last, a loan etymology from Mongolian /čuba/ ‘coat’ might be possible.
Considering the other five, a striking fact is that no less than three of these are Indo-Iranian loanwords (while only one IE loan occurred among the 26 words I suggest to have developed regularly: *śëlka ← *ǵalgo- ‘pole’):
- PIE *ḱm̥tóm > PII *ćata ‘100’
- PIE *ǵʰeslo- > PII *ȷ́ʰasra ‘1000’
- PIE *ḱerw- > PII *ćr̥va ‘horn’
I propose that these words, previously thought to have been loaned into Proto-Uralic (or the illusory ‘Proto-Finno-Ugric”), were actually loaned after the separation of Ugric, sufficiently late that in some ancestral stage of Mansi, *š was available as a substitute for the initial palatalized consonant. I actually have further arguments supporting this position for each of the words, to be detailed later.
As for ‘to dry’, I suspect late loaning from Komi /šural-/ ‘to dry’, which is from an unrelated PU root *šorwa-. The -l- formant, the unexpected short vowel in Sosva Mansi /surl-/, and the distribution in only Western and Northern Mansi all seem coherent with this explanation.
I have no ideas on what to do with ‘frost’.
Also left for later, for now: investigating the reflexation of *ś in word-internal positions, and in common Ob-Ugric vocabulary absent elsewhere in Uralic.
 West Uralic cognates such as Finnic *südän point to *ü, but East Uralic rather supports *e: cf. e.g. Hungarian szív, Khanty *sem, Selkup *siićə.
 Normally reconstructed as *śäkä, but actually only Finnic *säkä, *säkiä points to this: Mansi, Khanty (*seɣ) and Mordvinic (*śijə) all agree on *śekä. Mari *šij is ambiguous between these options.
 In light of Nganasan /sela/ this word may however instead rather belong among the words where *ä > *i.
 Proto-Mansi *ü in this and ‘to spit’ remains something of an anomaly, considering that I’ve proposed doing away with this PMs vowel altogether. At the Proto-Uralic level *ü is well established here though, and thus there is no problem in assuming that *śü > *sü. FWIW I’ve entertained the idea that perhaps the contrast between *s, *š was actually [sʷ], [s] at an earlier stage — but this doesn’t quite work with cases of *s next to the illabial PU *ë > PMs *ëë, such as ‘lightning’.
 Irregularly shortened. Finnish säie ~ Khanty *sööɣ- confirms the reconstruction with original *ä.
 Assuming palatalization only before illabial front vowels is not anything special: this has an exact parallel in Nganasan, where Proto-Samoyedic *ki, *kü are reflected as /si/, /ki/. The Finnic assibilation of *t to *c before *i but not *ü can also be compared here.
 An early assimilation *s-j > *ś-j might be a tempting assumption here, but this cannot be a regular development either, per *so/ëja > *tëëjət ‘sleeve’.