I'm kind of split on this, and it's actually an interesting question, so serious Kudos, Ajax, for posting.
The first thing this reminds me of is last year in English class. I had my hand resting on the desk while I worked, and the boy next to me (an acquaintance, not even a friend - the most we'd ever talked was when I helped him with his poetry - which he can not write to save his life. Still. Although that's not the point...) was apparently possessed with an urge to discover what my bracelet said. Reaching over, (this was the strange part, him grabbing my bracelet) he turned it over so he could read what the underside said.
Completely emberassed I pulled my hand away, admitting that yes, it was for diabetes, after he pegged it based of his friend who also has the disease. Thankfully, he was super nice about it and made a sort of complimenary-redirecting-the-attention comment/joke about it, which I appreciated. But I was still annoyed for him to have physically grabbed it, especially when I neither comfortable nor uncomfortable telling him about it. I agree it's one thing if people ask what it says; usually I'm free and open to answering that question, I've done it a million times before. But telling him was and is my choice, which he ignored.
So yes, I do mind if they do so without my permission. It's my diabetes, it's my bracelet - and it's my call whether or not I tell someone.