First off the [pilot of Air Force One][1] may or may not not be a commercial pilot (in the FAA sense) on account of the plane is operated by the military. For example the previous pilot of AFO [Colonel Mark W. Tillman][2] has no listing in the FAA database for pilot certificates so he does not cary any civilian privileges of flight although some might. 

Being in the service the pilot would carry all the privileges associated with his given rank at the time. 

As for other privileges there would be nothing stopping him from receiving a free meal at a local airport diner or the such but as far as I can find they have no special privileges anywhere.    

If you are asking what privileges they may have in the air that is a completely different question. They still must follow ATC procedures as any other plane in civilian airspace but the flight will undoubtably be treated in a certain way.  




  [1]: http://www.iloveairforceone.com/history/pilots/
  [2]: http://www.iloveairforceone.com/history/pilots/tillman/