I was looking through the Notifications list earlier in Setup > Email and discovered that it's possible to request a document to be uploaded to someone. This is a fantastic feature that because I always drag-and-drop files, I had no idea existed.
I suggest that the "Add Documents" button is a drop-down and has 'Upload from Computer' and 'Request a Document' as options -- otherwise a lot of people (I suspect) will have no idea that this feature is available!