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:
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.