[MunchkinMan]

Annihilation removes the card it is played on, always. If you play it on Mate, then Mate doesn't return to anyone; it goes in the box until you start a new game.

In case this is what's causing the confusion: You do not play Mate with another Monster card like you do Wandering Monster, since Mate is the Monster card, in a way. For example, if you play Mate on a Plutonium Dragon, you are now facing two Plutonium Dragons: The original, and its mate (represented by the Mate card), with a total combat strength of 40. If someone plays Ancient on the Plutonium Dragon (before or after Mate was played), you are now facing two Ancient Plutonium Dragons with a combat strength of 60!

Now, I explained above that we mildly insist that you get rid of the duplicating card, not the original, when it comes to Monster removing cards (except in the extremely rare cases of cards like Mommy), and that you only play Monster Enhancers on the original, since that makes life that much easier in solving the math.