[Inky]

In the original books, Hiccup becomes the greatest dragon trainer because he spends all his time watching dragons and learns to speak their language. A possibility. If you do the thing about the contract with the dragons, though, it might be better not to have any humans be able to talk to the dragons, at least not at first, or not well, so that why the dragons are leaving is a mystery to solve.

In the TV series (I haven't seen the film), some of the fun comes from all the dragons having different abilities - some breathe fire, some are just huge, some have armour-plating, some are electric, some can freeze things, some can breathe underwater, all sorts - and the riders having to MacGyver their way out of things using the powers of different dragons, their own or wild ones. (The one that can cut metal with its acid comes up a lot.)
If your kids like that part, then you'll need plenty of different dragons with different abilities to play with.
That probably doesn't mean you need to make lots of stat blocks separately. As Curmudgeon said, you can just have a few stat blocks, say "our dragon" "friend's dragon" "small dragon" "big dragon", and just change the superpower.
(If the impossibility of one little kingdom supporting dozens of whole species of dragon bothers you then they can be all the same species that's just really varied, maybe there's even some way to change how they turn out, as other people have said.)