As Suzy says - controller.
Two arguments for those who take the contrary view:
1. From B's perspective A does not need personal data. Although some aspects might be tricky to arrange it would be quite possible to get its staff trained anonymously using company credentials. "e.g. Please train three of our staff on Wednesday who we will provide with company passes and when completed give them each the evidence of completion to pass back to us.
2. From A's perspective it could not reasonably be acceptable to be at the mercy of A "Please delete all data relating to our personnel you have trained immediately". They have their own record keeping requirements, duties to trainees, and in many cases duties to professional and accrediting organisations. The reasons it needs personal data of trainees are its own - not B's as set out in point 1.
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