My insurance is what my employer provides me, so I didnt have to search. When we've had accidents in the past - motorcycle, auto, sporting (i play softball as well as ride among other things - and I'm a klutz!) Whether we've gone to the ER or the doctor after an accident, the medical insurance company has sent us a survey asking us if the injury was the result of an accident, and if so, can some other insurance cover the fees. My guess is that IF someone else was at fault, they, the insurance company, would go after that company. So far, it's never involved another party.
If you're looking for additional help in case of accidents, you might consider something like Aflac. Since we lead such active lives with a higher probability of serious injury, I carry aflac. It's relatively affordable and the pay back is pretty simple and quick. I smacked my hand in a virtual baseball game at Dave & Busters in a drunken stuper and thought I broke my knuckle. Went to the Dr two days later, paid my $20 copay, and Aflac paid me $120 for the "accident". I find it very useful since I'm prone to injury in sports - an extra $120 for spraining an ankle or dislocating a shoulder is very helpful for me