Waymo/Google is winning and has self-driving vehicles all over Chandler, Tempe and Gilbert Arizona at any given time.
For the most part it is going well, sometimes they act too safe but that isn't a bad idea for now, there have been a few complaints of Waymos going when fire engines or safety vehicles are on the road not slowing and in normal traffic slowing more than needed, or cars partially in a lane and slowing or stopping for them too much. I think the more grandma driving any self-driving is right now is good.
The biggest win of Waymo which is a competitive advantage and huge bonus is that by being Google/Alphabet, they also have access to tons of data and the best maps / map data i.e. speed limits, traffic, routes, dark/light changes to behavior, weather detection, data outside the CV/LIDAR that may be hugely beneficial. Other companies are going to have to pay for that or use Google, which they will collect immensely on in addition to use for free.
Think of that Tesla crash where other Teslas commonly had the same issue by drifting towards the ramp, Waymo/Google could correct that quicker it seems with their access. They need to be autonomous but also a neural network that can learn from other vehicles. They also need to tie into other systems like weather, day/night, emergency vehicles, traffic, routes etc etc. That Uber crash in Tempe, with day/night data and known pedestrian trails nearby + across the road, it may have been more cautious through that area and Waymo has that data with Google. Waymo is leading in multiple ways not just the self-driving part.