Ramp Controller reported some confusion and miscommunication occurred when three aircraft in adjacent gates attempted to push back simultaneously.

Narrative

Aircraft X had requested push off [Gate] XXX; a minute later [Aircraft Y on the adjacent gate; Gate XXY] had also requested push; seconds later [Aircraft Z on another adjacent gate; Gate XXZ] requested push. [Aircraft A from adjacent gate; Gate XXA] was already pushed so both Aircraft X and Y were blocked. I advised Aircraft X to push once Aircraft A had taxied away; hearing he had already received his taxi clearance from Ground; and seeing they requested first. I told Aircraft Y crew that once Aircraft X taxis away push approved. Mid push the push crew stops and proceeds to pull the heavy back into the gate; but then stops again to start the push sequence. At this point Aircraft Y Push Crew thought the Aircraft X was going back into the gate and proceeded with the push to which Aircraft X started pushing also. I stopped both crews (flight deck) and each explained the situation. Aircraft Y flight decks said the push back crew had faulty headsets which added to the confusion; and Aircraft X said it was a confused Trainee that had not pushed off that gate before; and again was confused by the Aircraft Y push.