My experience has been that the body which has the overarching responsibility to develop training forms, retain records, oversee the performance review system, etc. is often the Process Owner of Competence, Awareness, and Training.
That said, everyone else is responsible for following it, adhering to it, and conforming to it...and they may even have input into making it a better system in the future.
I've never been part of an organization with a Training team. Each group has been essentially responsible following the Training process (owned by HR) and providing the training that will hopefully lead to competent employees.