"Somebody" at your service!
Could someone please tell me, which of the sentences is correct?
They're both fine. You would certainly get them both marked correct on a Cambridge exam.
I've always thought people normally use personal object pronouns (me/you/her/him/it/us/them) in such sentences.
Yes, most people do.
I just read something and now am wondering... what if my whole life has been a lie and for some reason we need a possessive pronoun here?
Yes, the possessive is the more correct option. 'Making' is a gerund (otherwise known as a noun-like thing) so, strictly speaking, you need a possessive (which is an adjective-like thing/modifier). I use a possessive when I'm being careful and in writing. In more informal contexts, I'd probably use an object pronoun.
It's one of these cases where the 'wrong' form sounds more 'right' than the right one.