Actually, the messages part of the chatter pushdown should only show you the messages from other users. That's the display I fixed, but you'll need at least a second user to test.
The problem you see is because the new message was erroneously displayed immediately after posting, using the regular post template. I fixed that now. When you post a new message you will be redirected to the messages page, showing messages to and from the same user that you directed the message to.
Actually, the messages part of the chatter pushdown should only show you the messages from other users. That's the display I fixed, but you'll need at least a second user to test.
The problem you see is because the new message was erroneously displayed immediately after posting, using the regular post template. I fixed that now. When you post a new message you will be redirected to the messages page, showing messages to and from the same user that you directed the message to.