Hello,
The high level function of managing training is very often performed by HR, for example maintaining records and setting out minimum requirements as defined by policy, laws and established structure.
However, other departments certainly can and should take an active, responsible role in identifying and meeting training needs. This is in part because sometimes these needs are too technical in nature for a human resources expert to fully comprehend them and the various advantages/disadvantages of various contracted training providers. When this happens the departments should work in partnership with the HR department, to both ensure the records get properly updated and the entire system can become "smarter" based on the sum of its contributing parts.