Is "question" pronounced with an "s" or with an "sh" sound?

The consonant sequence /st͡ʃ/ in this word, and others, can indeed be replaced in pronunciation by something like [ʃt͡ʃ] as the result of assimilation. (I don't know if it would be correct to analyze this as representing a distinct phonemic sequence /⁠ʃt͡ʃ/.) This assimilation is not mandatory, and different people may assimilate at different rates. I don't have any data about that, however.

The people in the comments saying that this prounciation doesn't seem common in their area are not a particularly good source of evidence about the commonness of this pronunciation because many phonetic assimilations aren't easy for a native English speaker to notice: the sound [ʃt͡ʃ] in this context might be heard as /st͡ʃ/ by a native English listener, because that's what the listener expects to hear in the word "question".

Contexts where coda /s/ may become [ʃ] due to anticipatory assimilation

Coda /s/ in English may be assimilated to a sound like [ʃ] before a following fricative /ʃ/ or a following semivowel /j/ (Gimson's Pronunciation of English, Seventh Edition, by Alan Cruttenden, 9.4.5). These assimilations occur fairly often between words in phrases.

Certain words that originally contained a heterosyllabic sequence /sr/ may be pronounced with [ʃr], e.g. grocery ("Into the ears, through the head, out on the lips. How listening errors can trigger new instances of palatalisation" (Olivier Glain, 2013). To me at least, it feels like the pronunciation of grocery with [ʃ] contains the phoneme /ʃ/: it coexists for me with a variant pronunciation with /s/, and only /s/ is possible for me for some of the words where other speakers may have palatalization, such as anniversary.

Contexts where onset /s/ may become [ʃ] due to anticipatory assimilation

Assimilation of /s/ to a [ʃ]-like sound in sequences like /str/ is maybe a somewhat different phenomenon in that /str/ often occurs in English as a tautosyllabic cluster (e.g. in the word street). Tautosyllabic [ʃt͡ʃ] may occur for some British English speakers as an onset cluster in words like student due to assimilation. Likewise, the onset cluster /str/ may undergo assimilation, resulting in a cluster pronounced something like [ʃtɹ]. Glain cites a passage in Wells's LPD 2008 that mentions these two contexts where onset /s/ may undergo assimilation, as well as a 2011 paper by Rutter:

Rutter (2011) uses acoustic measurements to compare ten English speakers’ realisations of the onsets /ʃ/, /ʃɹ/, /ʃtɹ/, and /s/. He finds out that most of the occurrences of /str/ clusters produced by these speakers fall within their normal range for /ʃ/, as opposed to various intermediate phonetic realisations falling somewhere between a typical /ʃ/ and /s/. The results indicate that the change towards /ʃ/ is complete for those speakers.

So acoustic measurements at least may provide some support for writing /ʃtr/ and not just [ʃtɹ] for speakers who have assimilated onset /s/ in original /str/. I don't know of a similar acoustic study that covers words like question.

I haven't read the Rutter paper, but another paper that cites him says that

Rutter points out that in his own experiment, several of the sounds initially perceived as [s] were found under spectrographic scrutiny to more closely resemble [ʃ] (34)

("On The Perception of /s/ and /ʃ/: Considering the Effects of Phonotactics", by Kristen Scudieri, 2012)

John Lawler wrote in a comment:

It's quite normal for the /s/ in an /stʃ/ cluster to anticipate the upcoming palatal affricate by being palatalized itself to /ʃtʃ/. Consonant clusters reduce and assimilate, for the most part. Native speakers can't usually tell the difference between /tr/ and /tʃ/ before a resonant or a stressed vowel, for instance.

The reason that this happens is that the tongue has to retract to make the /tʃ/ affricate which has a post-alveolar place of articulation. In anticipation of this retracted place of articulation, the underlying /s/ gets moved backwards, thus changing it to /ʃ/.

John Wells's Longman Pronunciation Dictionary gives both /ˈkweʃtʃən/ and /'kwestjən/ as alternative standard pronunciations of question.