Those are people who have a "duty of secrecy". There are many ways to say that. In some contexts, people tend to use the word "secrecy", in some "confidentiality" is more common. Some people use "duty" more, some use the word "obligation". In officialese they often "are bound by" or "are subject to", rather than "have" the obligation.
Edit: you can also leave the "duty" or "obligation" out, and say bound by professional secrecy/confidentiality.
But I get the feeling you are looking for a term that includes a personal noun, is that right? You don't want to say "person who..."
A secrecy-duty-bearer, so to speak.
!! Please note: I would never say that, other than in jest.
Both "holder of secrets" and "bearer of secrets" sound acceptable to me, though a bit... old-fashioned or fantasy-fiction-fraught. But neither of them tells me that the person has an obligation to hold those secrets confidential. Only that they have secrets that they are keeping.
