New Client SBS2003 Exchange setup

I acquired a new client that is running a SBS2003 server. They host their email outside with a 3rd party which is really screwing up the email. I recommended that they bring their email in house. I have a simple question. The previous Tech setup the server using a .LOCAL domain. Their email addresses have a .com domain extension. Seems as though he setup email accounts on the server but never used them. The email address listed next to their profile says. user@mydomain.local. I went into the users profile and changed that to user@mydomain.com. I checked the settings on the server and they seem correct. I checked the SMTP Connector and the FQDN was computername.mydomain.local, Now I'm pretty sure this name does really matter. But I'm hoping that the server will accept emails coming from mydomain.com. Am I correct with this assumption?