Reinforcement Learning - Agent training
Ideally when training an agent you would reach an equilibrium between both player strategies strategies (hopefully a Nash equilibrium). When using only self-play this equilibrium might be different to the one you would reach when competing against a specific player, in particular, taking into consideration that the other player might not be very efficient, may have a different reward structure, etc.

Now, answering your specific question about the environment.. I would say that the critical aspect is the game being played. You should check if you are dealing with a zero-sum game, cooperative vs non-cooperative, etc.