As for LAN syncing, it needs to connect to the net to know when to sync, but then snycs locally:
Quote:
With LAN syncing, Dropbox will look for the new file on your Local Area Network first, bypassing the need to download the file from Dropbox servers, thus speeding up the syncing process considerably.
LAN sync is an extra advantage for use in locations where computers are networked together over the same router or other local area network.
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https://www.dropbox.com/help/137/en