Actually you should look up the info there, you actually don’t which is what a lot of fansubs rely on, they mostly only will own your translation if they chose to formally translate and publish commercially a translation in that country. If they don’t, you can distribute your translation for free. There is a lot of variability on this per country too, with very interesting laws in greece and germany in particular.

> If you write a book and I translate it to a different language you own the copyright on my translation.

Not quite true. You and I both own the copyright to your translation. Neither of us can publish it without the other's permission.