you said that it was the other party's fault. do you have any evidences? if you have strong case, you may claim with other party's insurance company too if you like. the key point is the other one does take the responsibility as admitted his/her fault and he/she has sufficient insurance coverage. otherwise, you may get troubles. the worst case, the other party deny fault and claim it against you or does not have sufficient insurance coverage, and you don't have other than collision coverage (comprehensive coverage). then, the other party's insurance company will pay nothing to you and your insurance company will do nothing for you too.
good luck.