I have a workstation running Ubuntu 14.04. And there's nullmailer trying to send emails (we are talking hundreds per day) that the sme8.1 is refusing. Then I kill the nullmailer on the workstation. I thought that sending emails as an interprocess to the various users/services of the workstation is an "essential" service.
1. I have no idea what is nullmailer is trying to send
Nope, nor do we !
http://wiki.linuxquestions.org/wiki/NullmailerBit of a contradiction there somewhere if you don't actually know what nullmailer is trying to send
"Interprocess" ? Who can tell. You need to find why it is installed and what it is doing.
2. Shouldn't the SME8.1 accept the message since it's from the inside network (even if it's from the root of the workstation) and therefore not a relay?
As per Stefano.
http://ubuntuforums.org/showthread.php?t=1488079 #1 Gives some details on auth, though you could have had a look yourself

Any idea what I can do to stop it?
Stop what ? nullmailer ? Whatever process that is generating huge amounts of emails ? SME refusing mails ?
You need to go back to square one and have a think through things a little more logically and provide some more detailed information after you have fully investigated.
Who installed nullmailer, and why ? Why are there 'hundreds of emails per day' and where are they going ? How do you know SME is refusing mails ?
Also remember there are a lot of SME users here, but no guarantees on how many use Ubuntu on the desktop. I do use Xubuntu 12.0x myself but have never heard of nullmailer until this came up. It sounds like the problem isn't with SME but your desktop.
HTH
B. Rgds
John