No one knows this structure.
You may not remember learning this rule about using the possessive with a gerund, but I assure you it was in an English grammar book you had at some point. It goes like this: some words ending in “ing” should be introduced with a possessive rather than an object pronoun. That’s the case when the “ing” word is a gerund, which is a noun.
To make this easy for you to understand, here are sentences that naturally seem to follow this construction correctly:
Their whispering disturbed everyone who was trying to watch the movie.
You probably don’t say, “Them whispering disturbed…..”
They took my arriving early as a sign of eager participation, which was not the case.
Again, it doesn’t sound right to say, “They took me arriving early….”
My constant apologizing only made them angrier.
Would you say, “Me constant apologizing….?” Unlikely.
His dancing was fun to watch.
Was “him dancing” fun to watch? We tend to say/write this correctly.
I think these examples sound correct because a word like “dancing” or “singing” is recognizable as a noun. But in many cases, using the possessive is not as intuitive, even though it is still correct:
He didn’t appreciate my commenting on his new facial hair.
vs. the error of:
He didn’t appreciate me commenting on his new facial hair.
I hope you don’t mind my interrupting you.
vs. the error of:
I hope you don’t mind me interrupting you.
To prevent my mom’s having to drive home late, I insisted she stay overnight.
vs. the error of:
To prevent my mom having to drive home late, I insisted she stay overnight.
You may have chosen the second choice in each example, but technically that would be incorrect. All three should take the possessive. So your ear is often an unreliable ally in this decision.
Sometimes It’s Not a Gerund
However, sometimes an “ing” word is not a gerund; it can be a participle. When it’s a participle, which is a verb form rather than a noun, you shouldn’t use the possessive:
I saw them still waiting for the bus after I’d circled the block twice.
I get annoyed at people claiming to be someone they’re not.
On the first example, your ear won’t let you write, “I saw their still waiting for the bus….” In that case, you can trust yourself to recognize that “waiting” is a participle.
On that second example, you could make an argument either way. Could “claiming” be a gerund and then it would be “people’s claiming”? I suppose so. But even highly educated readers may stop and have to look at that sentence if it has the possessive, and when you write a book, you don’t want to stop the reading fluidity.
Then there are sentences that require the possessive to be correct, but I’ll opt for making an error instead. Examples:
My basketball team’s winning the competition gave us all confidence as we prepared for the next challenge.
vs.
My basketball team winning the competition gave us all confidence as we prepared for the next challenge.
In some sentences, it’s important to figure out whether the “ing” word is a gerund or a participle, because using the possessive will change the meaning:
I wanted more info on the man juggling three jobs.
vs.
I wanted more info on the man’s juggling three jobs.
Did I want more info on the man, as the first sentence implies, or did I want more info on how to juggle three jobs?
Get It Write has a good explanation of this. In general, though, I am not judgy about it. If you want to drop that possessive, in most cases I’m okay with that decision. But as with so many grammar rules, I usually reword a questionable sentence. I would never write a sentence like the above “basketball” example. I’d say: Winning the competition gave everyone on my basketball team confidence as we prepared for the next challenge.
There are so many ways to convey the same thought that I try not to get trapped into these gritty choices.