What happens when someone posts on your profile page? Are you supposed to reply? Is this another way of sending a private message or opening a conversation? Whats the protocol ? What do I do? What do I do !?!? Yeah, I know...don't stress.
Profile posts are public. Your choice whether or not to respond. If you do not wish to receive these messages, there may be a way in your Profile to shut them off. In my position I cannot do so, thus have not checked for possibilities.
It is NOT private (as a conversation is), however most people will never see it. Think of it as something like your timeline in Facebook. If the post/discussion is something that would be of general interest, I would suggest moving it to a forum thread. If something private, move to a conversation.