"Their", in situations where strict grammar would indicate "his or her", seems to me a perfectly logical and convenient development of language.

I see it as analogous to the way that the formerly plural "you" has come to be accepted as the normal singular second person (resulting in a perceived need for a replacement plural second person, met by such terms as "yous" or "yez" or "y'all", and this remarkable thread.)

But I'm with you all the way, mmb, on the I - me issue, when it arises from a failed and misplaced attempt to sound correct. (Leaving aside the separate issue which arises when what is involved in using the "wrong" personal pronoun isn't that, but reflects a regional or dialect usage.)