It becomes a Private (or semi-private) chat.

Here is the use case.  You know person X is on the site/forum.  You send them an invite to join a chat.

PersonX is happily surfing the forum, and gets a notification on page refresh (Like PM) & asks them to join a chat.  That link brings them to the shout box page.

The shout box can be a 'party line', where if you happen to enter it when others are chatting.. you see their conversation.  Otherwise, the matrix of complexity would be too much for most users.