Well you'll want to limit it so only the one machine can access the database. You'd do this in your pg_hba.conf file. And use an encrypted connection if you are worried about other computers eves dropping.
Well, I still want the lan computer to be able to access the server database, retrieving records via queries. I just don't want it to be able to see ongoing queries/edits/updates on the database or specific tables. But that's perhaps impossible in the first hand?