@adavis444,
First off, we need to determine that the -ing word is used as a gerund, not as a present participle.
"I recall Bill asking, not you."
Here, if we take it to mean that Bill did the asking, whereas you did not, as far as I recall; then "asking" is a verb, the present participle of "to ask". Bill and you are the objects of the recollection (or non-recollection). "Asking" is a verb.
"I recall Bill asking, not yours." is wrong, in this sense.
Whereas
"I recall Bill's questioning, not yours."
If this is intended to mean that both Bill and you questioned someone, perhaps asking different questions, then the questioning is the object of the recollection, and is a noun; specifically a gerund. Here the possessive should be used. Likewise it is correct to say
"I recall Bill's asking, not yours."
which means something different to
"I recall Bill asking, not you."