Please run "mailq" to look at the mail queue and look into /var/log/mail.log. If the mails are not sent, there should be a reason.
Can you please try to send emails from and to two accounts on the same UCS system?
If you have to use an additional server or not depends on your Internet connection. If your server has a static public IP address and you can assign it a reverse DNS entry, it should work. If you cannot do that, then you should use your ISPs mail server as a relay for sending mail away. Receiving mails directly on your own server would still work.