The receivers would be recording more than the information contained in the GPS messages, they would be recording phase and signal strength data for all the satellites in the visible constellation.
Sophisticated post processing software would combine this information across multiple receivers, along with published satellite ephemeris data, to produce an accurate position solution.
Realtime positions cannot be that accurate due to affects like ionospheric refraction etc.
I used to work with a mobile system that recorded the GPS data along with inertial information (at 200Hz) that in realtime gave a solution that was usually accurate to within 30cm, and got to better than 10cm when combined with static ground station data in the post-processing step.
There are systems used in agriculture that are very accurate (10cm-ish) that use differential-GPS in realtime. The trick is your mobile unit has to be in constant communication with the differential ground station. Works ok for tractors, not so well for an aircraft 200-300km away. For differential-GPS to work well both units need to see the same satellites.
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