I assume, by then, you'll see massive insurance discounts if you provide the feeds to your insurance company. They'll use these feeds to evaluate your risk and the risk of every car around you. Cars, especially those with manual drivers, who are seen causing unsafe conditions will receive an eventual, or maybe immediate, notice: "Your premium has been increased for aggressive driving".
In countries where regulating things is understood as a purpose of government the availability of self-driving will end the "everybody needs to" justification for lax driving standards.
First thing you'll see is Continuing Education. Today for personal drivers testing is once and done. If you can hold in your instinct to aggressively cut off other drivers, pass on the inside and generally be a maniac for the length of the test you're set for life. There are small moves towards more continuing education (e.g. Speed Awareness requirements for people who keep getting tickets) but that'll speed up enormously once self-driving is viable.
Compare the regime for driving an articulated lorry (mandatory refresher courses, licenses automatically expire and you must be re-tested) to my grandfather being legal to drive long after he was in no physical or mental condition to be safe on the road.
Then I think you'll start to see tightening of basic requirements. That incautious fast turn you took becomes a Test Failure not a slap on the wrist. All fatalities result in lifetime disqualification. And when your lawyer says "My client needs a car..." the judge says "This isn't a license for a car, it's a license for driving. Buy a car that drives itself" much more often. Rich footballers who used to get away with this have already started to see judges say well, why don't you just hire a chauffeur? Self-driving tech would push this down to middle earners.
See, I come up with a tongue-in-cheek idea that with a creative could make for a great set of movies.
HighwAI or HAIghway or AI-95 or Night on the Interstate where the plucky action hero has to save the day with the help of an old car, a grizzled mechanic, and his sweetheart from a malignant swarm of AI controlled cars that have terrorized and paralyzed the country by killing any humans they sense near the roadway. Or a small town is terrorized by a swarm of autonomous vehicles that lurk the stretch of highway between them and the next town, and some disaster necessitates a massive human piloted convoy protected by the town sheriff and traffic cops as best they can until they can eventually make their way to a rendezvous with the State Patrol. Or teenagers having to avoid a gruesome death at the whims of autonomous vehicles on the way to Grandma's house.
Then you all have to bring insurance into it.
Seriously HN. Keep this up, and I don't think we can be friends anymore.
How that works will depend on whether the insurance company cares more about how you drive, or whether you cost them money in payouts. If there are no claims related to your policy, and haven't been for years, then it doesn't really matter how you drive.
In absence of competition they might use it as a way to gouge you for more money, but there is no lack of competition in the insurance space.