The traditional standard pronunciation is [sri], i.e. with a dental/alveolar allophone of the sibilant, similar to English s. That's what you will likely find in the older descriptions of the standard colloquial.
However, I am not sure how widespread this still is. In my own pronunciation, I believe, it is now more retracted, i.e. something like a retracted alveolar sibilant or an alveolo-palatal sibilant, which may sound somewhat in between the English s and sh sounds. My canonical Bengali sibilant (for example what I'd have word-initially before a vowel) varies from the palatal (similar to English sh) in the slow deliberate speech to more advanced tongue positions in more relaxed speech. It is possible (but hard to confirm for me) that in my relaxed speech I realize the sibilant more or less the same way whether there is an r after it or not. However, what I will never do is to pronounce শ্রী with a real palatal sibilant, like English sh, however slowly and deliberately I speak. That just doesn't sound right.
The normal pronunciation of শ is “sh”, but when a rô-phola (্র) is attached to it, it’s pronounced as “s”, meaning that শ্র is pronounced “srô” and not “shrô”, therefore, “sree” is the correct pronunciation for শ্রী. However, loanwords do not follow this rule.