1 domain name, 2 sites, no VPN. How do I configure Exchange?

I have one organization that just merged with another and we are trying to get email configured with the same .org address. One site is using Exchange, one is POP3-ing the domain server.

Both work well until we try to email eachother. Then the message “User Unknown” blocks communication.

Do I need to set up a VPN and put all addresses on the Exchange server or is there another way?

I saw something about setting the SMTP-Out in Exchange to an SMTP relay server instead of using DNS, but have no experience with that. Will something like that work? If so, how do I do it?