I agree with Lettuce with what they said. Also I think it makes it fun to allow people to plan and plot backstory. And I'm sure the Sinnoh government is pretty powerful, they can track down humans with legendaries fairly easily. They have spies, researchers, investigators they could pull it off. And like Lettuce was saying, it would feel more natural as if the trainers were picked by the Legendaries.
And these backstories between legendaries and trainers don't have to be too complicated. For example Noivian I was thinking Suicune could've met Nova near a frozen lake just outside of Snowpoint. Nova was training and Suicune being a strong empath sensed that she was having mental anguish with her sister, approached her in her human form and started trying to talk to her. Once Cressida got to know Nova, she would reveal herself as Suicune and would choose Nova to be her trainer since Cressida felt that Nova was a strong, but kind and caring person who only wants what's best for her sister and family.
Sounds good!!