For 2 player controllers:
1P Start button
1P Coin button (usually the R Bumper)
2P Coin button (usually the R Bumper)
2P Start button.

Played on Linux Mint 17.3 64bit with two Logitech F310 controllers set to XInput.

This is very much true. It works that way with Garou MOTW on Linux Mint. Probably the case for the versus and co-op modes of all of SNK's games. It's how the Neo Geo arcade games worked as well: the 2P had to load a coin in the 2P coin slot to get a credit to join in.