Doppler shift?!!! Where did you read that?!! They most certainly do not use any such method. They are pure positioning devices. Any other stats such as speed are calculated from changes in position over time. Think about what doppler shift is. It'd only be accurate if you were going directly towards or away from the satellite. For the most part, your gps device is using satellites which are nearly perpendicular to your direction of travel. They use differences in the timings received from different satellites, and as a consequence makes your gps handheld an extremely accurate clock. This is not the same as doppler shift.
Paul