B737 Captain reported almost contacting another aircraft on the ramp during push-back after another aircraft crew shouted over the radio for them to stop their push-back. Captain was able to have his crew stop and later learned that one of the wing-walkers was a 'fill-in' who may not have been trained. Communications with ground personnel were complicated by wireless headsets that didn't work necessitating using a wired headset by the push-back crew.

Narrative

Pushing back from Gate in ZZZ our aircraft was almost pushed into another Company aircraft that had pushed back from another gate. The Captain on the aircraft that pushed back from the other gate shouted on Ground frequency for our aircraft to stop our push. I quickly told our tug driver to stop the push which he did immediately. I then asked the other aircraft to come up Company frequency and let me know what was going on. He said we almost pushed into their aircraft and estimated we were about 10 ft. from his aircraft. We pulled forward so they could have room to taxi out. The wireless headset on our aircraft was not working. Our Ground Crew tried two headsets; neither worked so they found a backup wired headset. This apparently precluded them from being able to communicate with their Wing Walkers and reduced their visibility when pushing. I was informed of this after we debriefed the close call with the Ground Crew.The Ground Crew could not find a port Wing Walker. The Crew was short a Wing Walker and fortunately found someone who was willing to help. It's unclear to me if the person they found was trained or unsure how to communicate if they need to stop the push if an issue occurred. The aircraft we were almost pushed into was on the port side of our aircraft. Ground gave us push back clearance without a notice that there had been an aircraft that pushed off of the other gate. This might have been because it was some time earlier and they assumed it was gone. Excellent work by our Ground Crew to stop our aircraft when I instructed him to and the heads up from the Captain of the other aircraft avoided a near disaster or bent metal. 99% of the time ground will inform you that someone has pushed. This probably would have broken the chain of events. Headsets. Establish a time when Ground Crew can verify the headsets are working with the Cabin Crew. Also understand why they so often fail.Training for Wing Walkers. Often; I see Wing Walkers not even looking where we're going and at the ground or the wing tip. Training so they can understand their role and how they can prevent accidents. Training for Ground Crew and Pilots on communication if a tow bar breaks. The Ground Crew stopped us quickly per my request and then started to pull us as soon as he saw what happened. Fortunately; I stopped him; but I talked to my First Officer (F/O) afterwards that there is always a chance this could have sheared the tow bar pin with the sudden change. In this case; with no motors running and very limited time to react; the Ground Crew would have to inform me fast and I would have to gently but rapidly apply the brakes. I suggest there is training; communication brevity and a procedure on how the Ground Crew rapidly communicates a loss of the tow bar and what a Pilot should do about it.