It would depend. Most drivers have to have insurance just to get a license. So their insurance would cover any accidents. But it opens the door to someone suing you as the owner of record if there was a major accident. Example: my ex-Brother-in-Law, when he was 18, just out of high school, had an accident. His friend, in another car and who had been drinking, started effing around on the road as they were headed to the beach with a bunch of friends. Both cars were full. The friend started seeing how close he could get, trying to scare my BIL, and eventually bumped his car. The friend jerked the wheel to the right, slammed into the guard rail, bounced back slamming my BIL and all the kids in the car with him into on-coming traffic where they collided head-on with a guy that was driving 20 mph over the speed limit. Everyone in my BILs car was killed. Since he was driving and since he couldn’t protest, the entire accident was blamed on him. His insurance paid for a lot of the damages but there were proposed civil suits against my in-laws, who had co-signed on the loan for the car.
The point of that tragic story is that even if you are associated with the car even minimally, you could be targeted for a lawsuit if there is a bad accident. If it was just a fender bender, their insurance should be adequate to cover the damages and no one would care about you.