To go back to OP's question, if you are in an accident, and it's someone else's fault, yes, their insurance should pay. Ideally your insurance wil deal with their insurance and you don't have to do much. This is if you are dealing with a good insurance company.

There are a couple of scenarios, depending on how YOUR insurance operates.
1. Your insurance pays the bills and deals with going after the other guy, you pay nothing.
2. You pay the bills, submit them to your insurance for reimbursement who then goes after the other guy.
3. You try and get their insurance to pay directly. Remember it's the other guys insurances job to give you as little as possible.
4. You pay out of pocket and try to get the other guys' insurance to reimburse you.