Page 21 of the FAQ clarifies that the mage who controls the reanimate effect is considered to be summoning the creature that is getting reanimated (unless specified otherwise). In the codex we are told that the controller of a spell or object is the person who cast it. Thus, you do gain control of the creature when you reanimate it because you are the one who has cast it.
Now, the ownership of the creature does not change. Your opponent still gets it back when you're through with it. Although if it dies a second time during the game, it is obliterated rather than destroyed, so it does not go into a discard pile again.