Competition has a winner and loser because we choose to recognize those classes. Nobody ever got an award for 42nd place in anything, but it is a class distinction just as first, second, or third place.
Actually, having a winner and a loser is the definition of a competitve interaction. Without those "classes", it is not a competition. By definition for it to be a competiton, there has to be something you are competing for, food, a mate, a job, a trophy, bragging rights, whatever.
If I am doing some jiyu waza training and I ask my partner to attack strongly and try to counter my techniques, that doesn't mean I am competing with him. If I am sucessful at keeping him off or controlling him, I haven't won anything and he hasn't lost anything. Competition might make it possible to formalize and structure these interactions in some good ways, but it would also change the nature of the interaction, and in my opinion, the nature of that art itself.
BTW. This is not a value judgement of you or your dojo. The training you describe honestly sounds like a lot of fun.