One more thing ... users on your moodle control preferences for 'messages' in that they have an option to get them only in Moodle Private Messages or/and also get same via EMail.
So any user that has both options turned on, that means more processing by moodle ... Moodle Private Messages really doesn't use SMTP to send to itself.
Sent you a test message (to the address seen here in Moodle Forums) via my system ... see your address is gmail.com ...when you get it, 'Show Original' when viewing to see the full headers of the message. You will see where and how it originated as well as other mail servers the message traveled to get to your inbox.
Mail never has been straight from originating to any destination on any other mail service/mail server involved (cept for localhost).
And I'll say it again ... one size doesn't fit all! :|
'SoS', Ken