Whichever client is "running" the script that says "player" in it, then that will be THEM. So if Joe is running the script, player == Joe...and Mary runs the script too, then player == Mary.
If you want to reference other players, you would need to name them in the game editor. When you place a soldier in the editor and make it Playable, make sure to name it, like player1 or player2, etc. Then use these names to access those objects.
Doolittle