Prepositions (in/near/throughout/with reference to) often begin sentences.
"Mine" is a possessive pronoun, however, and I don't know of any rules about starting a sentence with one or not.
In reference to some of the comments in posts above:
I think "mine and Jenny's" sounds right because it would be OK in these sentences:
The car is mine and Jenny's. or simply
It's mine and Jenny's.
The
Harbrace College Handbook partly clears up our original problem with the dual ownership of that car by explaining when to use apostrophes:
"To indicate individual ownership, add the apostrophe and s to each name:
Joan's and Sam's apartments (each own individual apartments)
compare to
Joe and Betty's mail (indicating mail belonging jointly to Joe and Betty)."
But the other confusing part of the infamous
"It's my and Jenny's car" sentence is the choice of either
my or mine. "My" is an adjective and
modifies the word "car" (It's my car.) "Mine" is a pronoun and
replaces the word "car" (It's mine.)
So, it is my opinion that the correct form of the sentence is
"It's my and Jenny's car" but after spending so much time looking in grammar books trying to explain
why I will be sure never to use this sentence again in my life.