I can't answer whether the governments did question one another's accomplishments but I can say that they couldn't.

There was plenty of radio chatter from ground control to the spacecraft and any interested parties on the ground could get an idea of the range and position of those broadcasts. I would imagine any fakery would have been called out quickly.

No, but the Russians were concerned that the US would challenge the validity of Gargarin's flight as he ejected from the capsule rather than landing with it, for safety reasons. They kept this detail quiet for several years, releasing photos that implied he had landed in the capsule by showing him resting in a reclining seat beside the capsule as though he had just been removed from it.