In Phoenix you can as long as you live in a 200 square mile area comprising big parts of the Phoenix metro. And not just to the airport -- you can take it dinner, to the golf course, etc. SF and LA "coming soon".
I, no joke, encounter 4-5 of those white Jaguar driverless Waymo vehicles that are covered in sensors and cameras every day. They're all over my neighborhood. I believe (but havent confirmed) that normal folks can use them in this area too.