It is not necessary to play online for this trophy. It will pop as soon as you start either a multiplayer online game or a co-op game in hot-seat mode. One controler is enough to play this mode.

If you want to play online, be aware you will only be matched with people who have the exact version of the game you have (base or with any or all combinations of the several DLCs). If you have DLCs, these can be turned off under Downloadable Content from the Main Menu.

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FYI - This trophy's name is a wordplay of the term "LAN Party" which, before the wonders of the internet, was the only way people could play against each other. If you are not from that time, this meant having to create a Local Area Network (LAN) established between compatible devices which had to be fairly close to one another.