Air carrier Captain reported a near collision with a company aircraft during pushback at STL.

Narrative

Flight was late departing by more than two hours due to a ground stop in [destination] first; then by rain and lightning in the STL area. We finally had pushed back from the gate. It was dark and heavy rain. Because there was lightning in the area; crews had to use hand signals with ground teams. As our ground crew disconnected and returned to gate area; we noticed [an adjacent] gate began pushing back towards us. If you look at STL ramp; there is no room for both to push at the same time. I called on Company radio for them to stop while my First Officer turned on all of the aircraft landing lights. Luckily; the Ground Crew stopped pushing before there was an aircraft accident. We got them to shut down and pull forward so we could depart.I think there are a few things happening here that caused this almost aircraft collision to take place. One; overall visibility was very reduced from it being night; raining hard and the ramp area not well illuminated. Two; the use of hand signals during pushback reduces communication between Flight Crews and Ground Crews. Three; the fact we have been adding new Employees recently to fill shortages results in Employees with very little experience and possibly training. The aircraft on [the adjacent gate] should have never started to push. The Pilots in that aircraft can't see behind them so they rely on the Ground Crews to look and see if there is anything behind them. Kind of hard to miss a [large aircraft] even in the dark and rain. Their Wing Walker was giving the stop signal to the driver; with crossed wands and aircraft continued to push back. Only thing I can assume is Ground Crews are either not trained well or they are just not following their Ground Ops procedures. Not sure this Wing Walker concept is really being taken seriously by the required Ground Ops Personnel.