Moving files between shared Dropbox folders
Solution 1:
What's for certain is that Dropbox will not re-download files that already exist on your machine. So if the OS simply moves the file, but the actual file stays the same (probably based on a checksum), then Dropbox will simply move it in the index as well.
For other users of that shared folder, Dropbox will simply say "file has been added to your Dropbox" (or similar), but it won't download the file from the Dropbox servers. You could then, e.g. after a few minutes, safely delete the file from the original folder.
(This, according to the OP, definitely works on Windows. I will check Mac OS X later.)
Solution 2:
From experience: Moving / Renaming a file via Dropboxes own Web Interface will reliably just rename the file on the sync'ed machines. Moving a file one of the boxes can cause a full upload and download.