Does Rover.com run a background check on their sitters?

Yes and you can see the type of background check they had done by hovering over the badge next to their name.

When you see sitters listed in the search result, you see badges next to their names. The one which is a blue circle containing a white shield with a red checkmark inside indicates the sitter passed a background check.

Evidently it's not a requirement because I see quite a few search results without that badge. I assume that means they opted out of a background check, not that they failed. (But, I'm not sure how Rover would distinguish between the two. If I feared I would fail a background check I would opt out. So, something about this seems like a gap.).

Rover does some very basic checking before approving new sitters. However, having a background check performed by a third-party company is optional. If having a background check is a key concern, then you can search for sitters with this badge.