Does the opposing player score if you damage out one of your own objectives with card effects?

Yes, it only matters it was destroyed not by whom.

Thanks!

What about in a multiplayer game? Which opponent gets to score it if you're the one who destroyed it?

As per the multiplayer rules the opposing team chooses. Page 2. Also this only matters for the DS team. LS just needs 5 total objectives between them in multiplayer.