Had this problem on my exchange 2007 server (hub transport role) for a long time.
Some users could receive mail, but when trying to send using POP3 or IMAP outlook account they got '550 5.7.1 Client does not have permissions to send as this sender' error. They could send and receive emails fine when logged into OWA.
This problem seems to have something with SMTP authentication, the workaround I was using was letting those users to open relay on my server, but recently my server started sending spam so I was forced to look for solution to this problem once again (I've tried to resolve this problem before, but unsuccessfully).
I had configured send connector, also receive connector, but nothing seemed to work. Today browsing the web for answer I think I finally got it to work.
Following information found on the web I've opened Exchange Management Shell and ran this cmdlet:
get-receiveconnector "receive_connector_name" | add-adpermission -user AU -extendedrights ms-Exch-SMTP-Accept-Authoritative-Domain-Sender
Tried to send email again and it disappeared from the outbox.