Currently, that situation isn't possible, as self-driving cars still require an attentive driver who can take over at any time.

In the future? That's a hard question, and I'm sure however the legal system deals with it first will be wholly unfair and illogical. It'll be a battle between car owners, manufacturers, state and federal bodies, cities and counties, certification and safety agencies, etc.

Probably the same person that gets points when they cause an accident with their bicycle: i.e., nobody, because you don't get points for that.

Are we talking push bikes? At least in the UK you don't need a license for that anyway. I guess it's possible you wouldn't need a license for a self-driving car, but while regulations demand a fit driver who can control the vehicle that won't be the case.

My point is, I think regulations will change as technology improves. As transatlantic flights started, engines were unreliable, so regulators forced airlines to take awkward routes when crossing oceans (so that they would be 60 single-engine flying minutes away from a diversion airfield). Now that engines are more reliable, the regulations are significantly more relaxed, sometimes allowing 330 minutes to a diversion airport.

The point is, regulations are not set in stone. Companies can advance the state of the art, and ask the government to regulate less strictly.

As it stands now, self-driving cars are a research project, so we'd expect them to be regulated very strictly. As testing shows them to be safe, then we can relax the regulations. If self-driving cars without a supervising driver end up being safer than a normal car with a normal driver, it would make sense to not require a license. If an accident happens, it happens. Car accidents are nothing new.