I don't have permissions to send emails on my own behalf

It turns out that the users somehow created alternate credentials in Windows. Outlook would then try to use those credentials when sending as the user which would cause it to fail. Just go to the Control Panel, then to Credential Manager, and remove any credentials that relate to your

 http://community.spiceworks.com/topic/206858-i-don-t-have-permissions-to-send-emails-on-my-own-behalf