A woman hit my car last month, damage was 2700 bucks. It was her fault (illegal left turn, as per the police report). I asked her to contact her insurance company. I faxed them the written estimate. Then they asked me to bring the car to the adjuster for inspection. He cut me a check for 2300 after that rightaway. My insurance company was not involved at all.
My colleague had a minor accident - not his fault. he contacted his insurance company first. They contacted the guilty party's insurance company. Both the insurance companies fought it out for a while. My colleague got his car repaired, thought he had taken care of everything, so didn't follow up. On the next policy renewal, hi premiums went up steeply. When he contacted his insurance company, they said that THEY had to pay the other company, so it was noted as a claim against HIS policy, so his rating goes down, etc. etc.
You file a claim with your own insurance, your premiums are bound to go up. Its a very different world now, with insurance companies trying whatever they can to make another buck.