I do not use Pidgin or Thunderbird. If I try to remove these it says ubuntu-mate-desktop will be removed. Is this just a metapackage that can safely be removed? This is on 14.04.
It should be fine. I received the same warning before removing the Rythmbox media player via Software Centre. I wouldn't recommend totally purging anything though. May or may not be some critical dependencies buried.
Rest assured, I did not end up with a broken system 
Just don't remove any desktop or driver components. I installed the Pulse Audio Manager once to troubleshoot some stuff, and tried to remove it with a purge, which resulted in rendering my audio broken. If it really concerns you, give it a go in a VM 1st.
That’s good to know then. It just seems a little scary seeing that message. I thought using purge deletes any config files created by the package, not dependencies. I know of course this doesn’t affect any config files saved in /home. I am using ‘complete remove’ in Synaptic Package Manager so things may be a little different. I have done this in Xubuntu and removing packages such as those I mentioned does not prompt to remove any others.
It should be noted that the meta-package "ubuntu-mate-desktop" *(or any -desktop for that matter) is likely to be missing upgradable packages should you upgrade in the future. Here's a quote from another thread about the subject:
While it's safe to remove, it's usually a good idea to install the meta package again before performing an upgrade, and repeat the process after an upgrade completes. There is a more advanced solution however, here's the thread: