Go Daddy has a setting in it's email service that, by default, prevents emails being sent to addresses on the site domain for example: an email from some_address@mydomain.com can't be sent to another_address@mydomain.com
The solution depends on the server configuration, the GoDaddy help line should be able to fix it for you. Here is one example as a guide:
In cPanel go to Email -> MX Entry; under Email Routing, select “Remote Mail Exchanger” and click Change; then verify that there are “no MX records”.
Thanks to user Dalana for this information.
