Tug driver reported during pushback the aircraft wing was almost hit by the wing of an aircraft being pushed back at a nearby gate.

Narrative

Person A was being trained by me on push back training. We were on gate XX pish out flight ABCD to ZZZ1. Person A commences to push the aircraft into the alleyway he was making his turn. Wing walker Person B stopped us and informed us that Aircraft Y push back driver was pushing out their aircraft out [gate] XY and was not paying attention to us and his wing walker was in inadequate positioning to stop Aircraft Y he was in his blind spot. Due to the pushback tow angle on Aircraft Y. we were watching the Aircraft Y push back driver he would not stop. I Informed Person A to put the push back in park and to inform the caption to set his brakes. I got off the push back and went over to try the copilot's side to stop the Aircraft Y Pushback and wing walker to stop their plane; I was jumping up and down waving my arms to get his attention. The pushback driver for Aircraft Y made a hard turn did not stop both wings from Aircraft Y came inches from hitting our aircraft winglet. You could tell that the winglets almost made contact. If it was not for are wing walker Person B paying attention and stop Person A this could went really bad.