I think they definitely require GPS or some form of satellite navigation mechanism to work. I really don't think they can use the local built-in radar to position themselves. I believe until now they have taken it for granted that GPS will always work. Theoretically we can place ourselves in the map based on visual recognition and radar, but its much more difficult than using GPS, and I don't think any of the current implementation have thought of a fallback mechanism..
Given a map, and starting from a known position, it is perfectly possible to know where you are by continuously updating your position by knowing your direction and distance covered, corrected by recognizing features on the map as you reach them. When done from boats or airplanes, this is known as pilotage, but it is much easier on land, where there are no winds or currents, and even easier when restricted to a road network. I don't know whether any autonomous car is equipped to do this, but it would seem to be one of the lesser technical problems of the field.
I believe it will be difficult in land than it is in air or water. For e.g., how dependable would be the the recognizable features, especially in lesser known roads. Without that if the vehicle narrowly misses a road (could be due to a new road coming up or slight mismatch in the map or a temporary diversion) then the direction and distance would quickly become useless as it can't place itself anymore in the maps. Human beings can use their intuition or ask someone to correct themselves, but it would be very sophisticated to implement on a self driving machine.
It is not correct to say that your displacement quickly becomes useless once you encounter some discrepancy in your map. It remains valid as a good estimate of your position, allowing one to recover from map errors so long as they are not pervasive. This is not a trivial programming exercise, but nor is it as difficult as many other aspects of vehicle automation.
