It reminds me of something that occurred a few days ago. I’m in the process of migrating a forum.
I granted administrator privileges to the admin of the old forum.
He’s a Discourse user on two other forums, and also a moderator. So he knows a bit about Discourse interface and navigation.
From his admin account (I didn’t write any guideline to him, I let him discover stuff) he told me he was surprised to see he could access direct messages from other people; He clicked a bit randomly on a public profile page and saw a list of direct messages, which surprised him.
He told me he suddenly thought “Oh well, it seems I’m somewhere where I shouldn’t be” (to be understood as: he shouldn’t be here if he has no administrative/moderation purpose at this time).
He has no idea that he could access these messages from the interface as easily as you access your own direct messages from your profile.
And it also happened to me a few weeks ago also, despite the fact that I’ve been an administrator of two Discourse forums since 2018… 
In my opinion, on the “direct message” tab from a user’s public profile, as an admin, there should be an icon or any kind of warning that clicking it is a moderation or administration action, as you’ll see things meant to be “private” (in 99% of users’ minds anyway). 