The company formerly known as Google (they created a holding company called Alphabet under which Google proper and other divisions are hosted, but it's hard to bring myself to call them Alphabet) is apparently about to kick its self-driving car project into high gear. Anonymous sources who are in the know claim that in 2016, the driverless cars will get their own company under the Alphabet umbrella, and will start to offer rides-for-hire services. At first, these are expected to be limited to "confined areas" like "college campuses, military bases or corporate office parks," but the goal is without doubt to graduate to public roads as soon as feasible, both from technical and legal points of view.
The Google electric and hybrid driverless cars have driven over a million miles so far, most of which are concentrated in the San Francisco and Austin areas, so it's likely that these are the places were a ride-sharing service would start.
If Google decides to take this approach and rent out a fleet of driverless cars by-the-ride, it will likely be competing against both Uber and Tesla.
While there are concerns that driverless cars could increase miles driven and traffic, I think it's a little early to tell. There are many variables at play. I think it's probably just as likely that large fleets of electric driverless cars that one can summon with a smartphone will probably make the rate of private car ownership plummet. These vehicles can spend all day get people where they are going rather than spend 95% of the time parked, like most private cars, so one of them can probably replace dozens and dozens of vehicles.
If most people (especially in urban areas) stop owning cars because these on-demand cars-as-a-service are more convenient and cheaper, millions and millions of cars would be taken off the road. Fewer parking spots would be needed, freeing up space for bike lanes, etc. Smart investments in transit could mean that for commuting, most people would use mass transit (as they should -- it's so much better to move a large number of people in a short amount of time), and for those trips that aren't convenient with transit, hiring a ride would be the primary choice.