The change will happen slowly, organically, over time. A self driving car will behave statistically as a very safe driver. Ownership of a self driving car should bestow upon you lower insurance rates. If the current insurance companies balk at the idea, the private market will take over and "self driving only" insurance companies will gladly take their place. Eventually, as more and more share of vehicles are self driving the size of the insurance industry will shrink significantly.
I see no reason to change the liability burden away from the "Driver". It may seem counter intuitive, but you are gaining economic advantage by using your self driving car. For that advantage, you accept the risks, and insure yourself against them. That said, operating a self driving car will/should carry significantly less risk and liability then driving yourself around does now.
That does not mean that the car makers are off the hook. Just like today, if a vehicle mechanically malfunctions in a way that the car maker is found responsible, the insurance company may attempt to subrogate the claim to them.