Before an -ing form (like talking), do you need the possessive form of the personal pronoun (my, your, his, its, our or their) or the object form (me, you, him, it, us or them)?
It depends! Both are correct. However, the possessive pronoun is more formal, and more likely to be used in writing. The object form is quite common in speech.
Similar examples are:
I hope you don’t mind my/me interrupting you.
He saw my/me joining the club as a threat.
The likelihood of his/him being there is remote.
I really appreciate your/you going to all the trouble.
Very occasionally, there may be a slight difference in meaning. For example:
They didn’t like me singing (They didn’t like the fact that I sang)
They didn’t like my singing (They didn’t like the fact that I sang OR They didn’t like the way I sang)
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