Captured monsters may be done with as the other character pleases, however, don't expect them to immediately warm up to you or beg to become yours, some arn't even capturable without a large enough party or arsenal etc, if you do manage to capture a monster, you may keep them, use them, kill them etc so long as the gory bits are done off screen or kept vivid.
However, that being said, capturing isn't required if you can appeal to their nature, each one has different personalities, therefore different ways to perhaps persuade, manipulate, or even befriend if possible.
Although be careful, or the hunter could become the hunted...
Some examples of defeated or captured monsters.