There were insurance requirements according to this archived https://www.nytimes.com/1989/08/27/travel/behind-the-wheel-in-the-soviet-union.html. It doesn’t really go too in depth about how the insurance works, just that it is a requirement. I am pretty sure it was state owned because pretty much everything was, but I couldn’t find a super credible source on it beyond people online just saying that it was state owned.

It was extremely hard to get permission to own a vehicle, and it was even more difficult to get gas and parts to maintain the vehicle. Usually only Soviet officials and well connected people had cars. It was such an issue that normal citizens couldn’t get cars there was a comedic satire film called Beware of the Car where an insurance agent would steal cars from politicians and scammers and sell the cars to give donations to orphanages.