Dropbox automatically performs a LAN sync if your computers are on the same subnet (they usually are if you're connected to the one router), so it's much faster than each computer downloading the content from the Dropbox servers.
https://www.dropbox.com/help/137/en
NAS units are fantastic, but the cheap ones aren't fast (compared to plugging the drive directly into a computer).
If one of your computers has a fast processor with plenty of RAM, you may get better performance by setting one of them up as a file server.