Should people in Pokémon lore consider legendaries/mythicals as "protected species"? Like, they could make it illegal to catch them or something. I personally don't think it's okay for anyone to own such a powerhouse Pokémon--it could be like owning nukes.
Now, there could be instances where one works with a legendary to fend-off a greater threat, or to help them protect their territory. In which case, it could be a temporary partnership. Or, you could get a license to research legendaries, putting you within legal grounds to handle them.
On top of that, legendaries/mythicals might as well be immortal, so they tend to outlive their captors/handlers and remain elusive to a point. Could be a public database that shows which person is currently handling/studying a particular legendary.

TL;DR -- Maybe no one should own legendaries/mythicals because ethics and semantics.