I was in an accident 5 years ago -- other driver's fault. I did contact my insurance and let them handle it and then subrogate.
The other night, my teen was in an accident in a car we don't have collision coverage on (much older car that we didn't think was worth the additional monthly premium), so we couldn't go through our insurance. Fortunately, the other driver (who is at fault) has valid insurance and I'm just dealing with them directly. I notified the claims department of State Farm (our insurance) just to make sure I wasn't missing anything or if I was required to notify them, but they told me I wasn't. They did tell me to call my local agent should I run into any problems with the other guy's insurance (Progressive). There is a police report and the other guy is obviously at fault, so I'm keeping my fingers crossed that I don't receive any type of pushback.