Thanks for your reply, Krileon but, unfortunately, this not only doesn't solve the problem but, also, is unclear to me.
So, let's say that there are two users, x & y and x sends a PM to y. Then, y can go to user x's profile (or indeed to any other user's profile), where there is a CB Menu item called 'Messages'.
One of the sub-menu items is 'Show Private Messages'. If y goes there, they get a 'Compose' dialogue but also above it, a menu with items ''Inbox', 'Outbox' 'Trashcan, 'Contacts'. 'Compose' and 'Forum'. Clicking on 'Inbox' shows a list of messages, which would include the message that y had sent.
However, this would be a very user-unfriendly way of determining if messages had been received and what those messages are. As some part of CB does know that a message has been received, there must be some way of using an autoaction to hook into this event.
How else would this work? Is there no way, other to that described, above, for a user to see that they have received a message and to go directly to their inbox? These messages don't even seem to go into the Activity timeline unless I'm missing something.