After some investigation I found that this was dependent on which view the user logged in with: Desktop or Content Editor.
The problem turns out that a Droptree with an empty source is going to be relative to the users starting position in the content tree. When users were logging into the Content Editor view, their starting position is set to /sitecore/content/ by default. I was able to give them access to the Media Library in through a Droptree by setting the field's source to "/" which gives them access to everything under /sitecore/
I hope this helps someone else.