Unfortunately, the ultimate answer is that it depends on the pet.
In most cases, yes, it is *possible* to get the other person's pet type. Always remember that every breed attempt is a crap shoot and and try not to bet beyond your character's means.
You will never get one of the 'rare' pets that KI has on the list at the top of this forum. They have been effectively removed from the breeding pool.
You may also have noticed an announcement at the top of this board letting you know that you will *never* get a pet you can't equip. If your fire wizard tries to breed his fire bat with a friend's satyr, he's only ever going to get a fire bat.
(That fire bat might get something cool like pip o plenty or spritely, but that's another matter.)