In terms of your connection, LAN and WAN isn’t really going to be the way to go except for amet very specific scenarios.

The Steam Deck and almost all multiplayer games connect via Internet servers and your steam accounts. Some games you can host and your friend connects directly via the Internet - games are designed to support that so you often don’t need to resort to local LAN/WiFi play.

It doesn’t matter that you’re next door to each other - you might as well be miles away from each other for all it matters - you both just need good stable internet connections to the remote servers, with decent speed and your Internet routers not too restrictive on your connections (firewalls not blocking access, relevant ports open).

So basically ensure you have a good WiFi connection. Even better you can also get USB c ethernet dongles or a dock for your steam deck with ethernet to connect to your router directly and avoid WiFi.

I play with my steam deck docked under the TV, ethernet connection to my router, hdmi to my TV and an xbox controller and Bluetooth headphones. I play on the couch with all the even fits of the steam deck.

Both of you docking your steam decks with ethernet connections to your Internet routers may give you a better experience.