I agree 'She and Mary' is correct as the subject in formal written standard English. But on MN, the style is conversational and certainly not formal, and like spoken English 'Her and Mary' as the subject is surely quite normal.
Certainly, if a colleague came up to me and said "Me and John are going down the pub - d'you fancy coming?" that seems perfectly grammatical for colloquial speech.
DadDadDad's suggestion may be colloquial but it is neither grammatically correct nor polite. Assuming politeness trumps proper English, at the very least it should be "John and me are ...". I suspect however that is harder to say and might actually lead the speaker back to the proper form.
I've always understood grammar to mean the rules of syntactic construction that competent users of the language follow.
"Me and John are..." is common enough in spoken colloquial English for it to be perfectly grammatical in colloquial English, as far as I am concerned. That doesn't mean it's grammatical in formal written English, but that's the distinction I'm making.
I am not sure competent usage can be defined by common usage. Indeed I would deny that the two can be equated for to do so is to invite reduction of the use of the language to constantly lowering common denominator. It is a very small step from spoken usage becoming written usage - how often do we see "of" used in place of "have" and"different to" in place of "different from" now being used even in what were once broadsheet newspapers? These usages have, I suggest, migrated from colloquial, but, in my view, incorrect, speech.