Conversations are private already. Conversations are available from the Inbox at the top right of the forum.
What you are looking at on Profile Pages are Profile Posts. The initial visibility of these can be set in User Registration Options in the ACP. However after that it's entirely up to individuals to decide on the visibility - whether they are public, members only, followed members only or nobody.
If users want to chat privately, they will use Conversations. Profile posts though are useful for posting stuff that wouldn't necessarily go into a forum post but does not need to be completely private (so in that sense it's a bit like posts on Twitter or Facebook).
It is very strange indeed.
Normally I use Firefox for everything and when I used the browser Opera to go to my forum and when clicking on a name I first saw the complete conversations I held with the member.
That is why I posted this thread; when I do the same with Opera now, then there are no conversations in the profile post.