There will be a huge public to-and-fro about negligence versus freedom, with driving manual being the new driving drunk, but the insurance premiums will settle it.
I predict that many in bigger cities will give up on owning their own car, and join car sharing services.
Then you can choose between a private car just for you, or a shared ride where the car optimizes routes and picks up multiple people.
Other benefits: it's just you? take a small 2 person car. On the way to a party with some friends? Get a nice Mercedes to drive you there. Need to transport some furniture? Get a pickup.
Of course this only holds for cities and densely populated suburban regions. Or if you don't have to drive long routes on a regular basis.
It's worth mentioning - which I already hinted at in another comment - that self-driving cars, awesome as this is, will be yet another area where our privacy gets compromised, same as with mobile phones. This will push governments to favour it in the long run and eventually outlaw "regular" cars altogether as too anarchic (or make using them so expensive and cumbersome that it's going to amount to the same thing).
When they truly understand that it's going to turn them all into taxi makers, that owning a car is completely passé, they'll panic. That will be their DVD -> streaming moment, but it will already be too late.
