Local domains get: 550 relay not permitted, authentication required

If you are trying to send email to local domains on your server, but exim returns this error:

550 relay not permitted, authentication required

it means that exim doesn't think the domain is local.  This is controlled by the file:

/etc/virtual/domains

so your domain is likely just missing from that file.  Add the domain to that file to make the domain local.

This is related to this feature:
Using an external Mailserver

where the User has the ability to un-check the "Local Mailserver" option, causing the domain to be removed from the domains file.