You add ancestors for whom you have sources to say they belong. You don't need to be a manager of the other profile to add parents to it, but it is considered good Wikitree practice to communicate with the other profile's manager when making what are considered major changes, such as adding parents. (This may not always be possible (inactive PMs are a thing) -- and oftentimes we get so caught up in "getting it done", we forget.)
No-one owns profiles on Wikitree, we manage them. There aren't separate trees, just - as you state - our own branches of the one Tree, where we try to attach our lines to the main trunk.