An Arkansas Times editorial said "Regardless of your personal belief, I would hope that you shutting up about your personal belief and letting me decide these things is something we can all agree to." Richard W. Chapman finds fault.
"The gerund 'shutting up ... letting me decide these things' is treated as a noun and thus requires the possessive form of the preceding pronoun," Chapman writes. "The sentence should read 'I would hope that YOUR shutting up ...' ."
A shade-tree grammarian at best, I usually try to stay out of the minefield of gerunds, genitives and fused participles. But if we must, we must. We'll eschew technical terms as much as possible.
There is indeed an old rule that requires the use of the possessive in sentences such as the one from the editorial. Some people take the rule seriously; some less so. One prominent grammarian has said "It's a niggling point, but one on which many people niggle." My attitude is the same as my attitude toward the old rules about not splitting infinitives and not ending sentences with prepositions. I don't go looking for occasions to break the rules, but if I think a sentence sounds and reads better with the infinitive split and the preposition at the end, that's the way I do it. Jane Austen was a great writer, and a grammatically correct one, but who today would write, as she did, "Elinor was prevented from making any reply to this ... by the door's being thrown open, the servant's announcing Mr. Ferrars, and Edward's immediately walking in"? This, on the other hand, sounds right, even if ungrammatical: "The rage and uproar over me becoming a Muslim was still at a fever pitch." That's from "The Greatest," by Muhammad Ali. If I have to fight, I'd rather fight Jane Austen than Muhammad Ali.
Maybe the Times editorialist shares my view. Maybe he didn't want to put two "your"s so close together. Maybe he dozed off while writing. Editorials can do that.
People who want more on this subject should check the "Fused Participles" entry in Garner's Modern American Usage. More respectful of the old rule than I, and knowing far more about it, Garner says, in part, "Especially in formal prose, the possessive ought to be used whenever it is not unidiomatic or unnatural."