Legendaries should be unique and not everyone should have one so I thought for a while and made a concept.
There will only be ONE of each Legendary either distributed by events or putting them in a specific location. Once a player captures the legendary, he has an X amount of time that he can keep it(Let's say a week). After that, he must battle other trainers in order to keep it. If he loses a Y amount of battles he loses the legendary and the ownership is tranferred to one of the winners(This section needs improvement so tell me your ideas). Also, if the owner becomes inactive the legendary is sent back to its previous owner or randomly given to somebody(This also needs improvements). There should also be a minimum amount of battles per month(Let's say 20) the trainer must do or he loses ownership of it. Finally, everyone who got his hands on a legendary at least once should get a medal or title.
Possible Problems:
1)Players defeating eachother to keep the legendary for themselves.
This can be easily fixed by making a list of the players who challenge the player and then randomize it. Also we can use the banhammer :P
2)Legendary getting stuck in the same person for a large amount of time.
This is not a problem because if he can keep it for that much it means he deserves it.
That's it I will add more in the future(sorry for my English).