With self driving cars, we still have to maintain enough cars to meet peak demand. Therefore there are going to be many cars that spend most of their time idle. It will be less than current levels but not by much (a good estimate will be it will reduce by the average number of cars parked during peak hours).
True, we still have to meet peak demand, but I disagree that it won't be much less than current levels. For multiple reasons.
First, once we get the human-operated vehicles off the highway, we can greatly increase highway speeds. Tightly-packed "trains" of automated vehicles can cruise along at, say, 100 mph. Automated vehicles, particularly with radio frequency communications for coordination, will also not be subject to the typical rush-hour slowdowns; as density increases human drivers have to slow down, especially to manage entrance and exit ramps. Automated vehicles in constant communication with surrounding vehicles wouldn't need that. This means that a vehicle that currently makes a single journey during rush hour may make two or three, exploiting the small amount of spread in arrival times that does exist.
During the interim period when highways are shared with human-operated vehicles, we can still get some of the same effect by partitioning of automated-only lanes, separated with barriers and moving at much higher speeds. This is only feasible where there are already at least three, and preferably four, lanes of traffic, but many heavily-commuted corridors meet that requirement.
Second, and even before removing human-operated vehicles from the highway, self-driving cars make mass transit much more effective and accessible in sub-urban areas. Some regions have large parking lots near transit stations in an attempt to facilitate the personal vehicle to mass-transit transition, but that's expensive (for the transit system) and not always feasible. Given self-driving cars, self-driving buses and systems that know the actual location of the buses, it will be a simple matter to have a car pick you up and drive you to the bus stop (or, even better, train stop), arriving scant minutes before the transit vehicle.
Self-driving vehicles also facilitate smaller, more independent, transit vehicles. Rather than building a system around 40-seat buses, you can build it around 10-seat vans. When you have to pay a human driver, going bigger is far more efficient. Without a driver, smaller vehicles are still more expensive than big ones on a passenger-mile basis, but only if both are running full. Given how often buses are mostly empty, a larger fleet of smaller vehicles will be more cost-effective, and with automated routing and automated electronic interaction with future passengers can be much more efficient. Consider the opportunities for efficiency if passengers' personal digital assistants (e.g. Google Now) notify the system when and where the passenger is intending to board the van.
Third, right now the cost of vehicle transit is mostly independent of the time of day. It costs a little more to drive to work during rush hour because traffic results in higher fuel costs, but not much, and people don't think about it at all. But I expect that rush-hour transit in automated car service vehicles will be subject to "surge" pricing, because the level of peak demand translates directly into the single highest cost for such a service: vehicles that sit idle much of the day. Of course, the surge pricing will still be lower than current personally-owned-vehicle transportation costs, but because it will be so visible and easily measurable I expect it will provide a stronger incentive for employers to allow their employees to stagger their arrival times, spreading the surge and reducing peak usage.
On that last point, it's worth noting that most employers of highly-paid employees already allow them considerable latitude in arrival times, and many employees take advantage of that flexibility to reduce the time cost of commuting. Lower-paid employees tend to be required to punch the clock, and their time is considered less valuable, so employers are less receptive to complaints about transit times... but might be more receptive to complaints about transit surge pricing.