What does happen when a supersonic aircraft passes another supersonic aircraft, regarding sonic booms?
When supersonic aircraft flying opposite routes cross each other, do pilots hear a sonic boom?


If the aircraft are near enough, yes.


  If they fly in the same direction, with the appropriate offset, does one hear a permanent sonic boom?


The boom will not be permanent i.e. static, but yes, it will be persistent. However, it should be noted that the shocks from various parts of the aircraft body will interact nonlinearly (mostly coalsce), resulting in unpredictable results. 


  If two aircraft fly in opposite directions at Mach 0.6, can they create a sonic boom by adding their individual shockwave speeds?


No, sonic boom is caused by the object travelling faster than sound while the disturbances travel in the medium with speed of sound and get 'bunched up'. So, the aircraft has to fly supersonic with respect to the medium


  What are the effects on the aircraft?


Shock wave is basically a pressure discontinuity. So the effect will be similar to that of an impact, whose strength depends on the shock wave.


  Is there a distance after which the effects become insignificant?


Yes, but it depends on the aircraft size and the speed. The shock wave produced by the bullet loses its strength in a short distance, while the Concorde has to be flown over sea and Tu-144 at over 35000' to mitigate the effects.