-I resented Bob's kissing my girl at the party
-I resented Bob kissing my girl at the party
What’s difference between the two sentences?
I'm glad that you are so conscious of the need for the possessive pronoun before a verbal noun.
The correct sentence is:
I resented Bob's kissing my girl at the party.
I do not 'resent' Bob ( I cannot resent a person, only something a person says or does - "resent" never takes a personal object). Bob may be a nice person and a really good friend... it was the 'kissing of my girl' that I resented, and it was Bob who did the kissing : Bob's kissing my girl!
Both sentences mean the same thing and are acceptable, although some traditionalists insist on the use of the possessive.
Without the possessive, the entire phrase "Bob kissing my girl" is viewed as a single object: the action of Bob [in] kissing my girl. In this sentence, "kissing" is a present participle.
With the possessive, the object is "kissing", and it belongs to Bob. In this analysis, "kissing" is a gerund. One objection to this construction is that actions are not things which can be owned, but are performed by a subject or some other actor.
When a verb takes a gerund as a verb complement --as do verbs like avoid, finish, not mind and resent)--, if the subject of this complement is different than the subject of the main clause --I don't mind (Bob waiting | Bob's waiting) here--, because the gerund is considered to be a noun-like form, to modify it one should use an adjective form (his, Bob's, etc.) rather than a pronoun form (him, Bob, etc.). At any rate, this is the rationale for the possessive's use here.
However, Google shows:
96 for "don't mind him doing"
9 for "don't mind his doing"
So modern usage seems to have dispensed with this distinction.