When you mean "local only" do you mean no Internet mail, just local accounts?
Try installing the mailutils package, then send a test message from the command prompt like this:
replacing "someone" with a valid account on the box. Does the message get delivered to /var/mail/someone?
echo 'this is a test' | mail -s 'test message' someone
Are you trying to connect with Thunderbird on the same machine that runs Postfix and dovecot or from another machine on the network? If the latter, you'll need to remove the restrictions in both programs that limit them to listening on only the localhost interface. In dovecot, you need to have a "listen = *" directive. In Postfix, read this discussion of the mynetworks directive.