I believe the point of no private or matchmaking is because the emphasis is to form a #FireTeam and in any mode you will be with your fire team of three. which is the number of classes making it a sort of Trinity. building a bond, strategy and synergy with your 3 #FireTeam members in all modes.
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you should be able to matchmake or make your own group. As it stands there are no tools for socializing in game so unless you have friends that play the game at the same time as you it's pretty hard to LFG. In my opinion this is a pretty broken game mechanic and almost kills the part of this game that was supposed to be revolutionary.