I don't think any of those have private servers that you can run and host games in. They're all far too dependent on microtransactions and in-game-purchased items to implement that kind of robust and long lasting system. They will die because of this as soon as they are not profitable.

Oh, and Overwatch is dead and unplayable. Blizzard unilaterally killed it despite people wanting to keep playing. There is overwatch 2 but that is not the same game.