The other day I happened to click on someone's name by accident. I was using TB from my phone screen where everything is smaller! Anyway, I was able to see messages that were sent from other TBers to that person, and I assume that they were meant to be private.
I just clicked on someone else's new profile picture to have a better look (!) - and again I was taken to messages.
I have just clicked on my own name which takes me to messages (not my inbox) and all I can see is 'X sent you a message'. I don't know if others can actually read those messages...
Anyway, it would be useful to know how to clear them. Maybe you should all have a look at what's visible on yours too!