***SOLVED***

Two PCs both with
Xubuntu 22.04 with Nvidia 525 driver

The game runs correctly but neither PC can create a LAN game and have it seen by the other.
Internet multiplayer works correctly.

It looks as if a server can't be started from inside the game but relies on a separate process. The windows version apparently needs a .bat file to be run in the background on one of the PCs. So probably on Linux some command must be run in the terminal.

I couldn't find any documentation. There is a shell script in the Overload/game folder

RunDedicatedServer_x64.sh
or
RunDedicatedServer_x86.sh

So it is necessary to open a console at this directory location, and do

sh ./RunDedicatedServer_x64.sh

It returns a long list of error messages which were too long to paste below. But it seems to work despite this.
I haven't narrowed down yet if it requires Steam to be running in the background, as the error seems to say.
The terminal window should be left open.

in the below I've replaced the PC-specific paths as [INSTALL PATH] and [HOME PATH]

==

Overload/game$ sh ./RunDedicatedServer_x64.sh
Set current directory to [INSTALL PATH]/Overload/game
Found path: [INSTALL PATH] Overload/game/Overload.x86_64
Mono path[0] = '[INSTALL PATH]Overload/game/Overload_Data/Managed'
Mono config path = '[INSTALL PATH]/Overload/game/Overload_Data/Mono/etc'
Preloaded 'ScreenSelector.so'
Preloaded 'libCSteamworks.so'
Preloaded 'libopenvr_api.so'
Preloaded 'libsteam_api.so'
Unable to preload the following plugins:
libUnetEncryption.so
ScreenSelector.so
libCSteamworks.so
libsteam_api.so
Logging to [HOME PATH]config/unity3d/Revival/Overload/Player.log
[S_API FAIL] SteamAPI_Init() failed; SteamAPI_IsSteamRunning() failed.
[S_API FAIL] SteamAPI_Init() failed; unable to locate a running instance of Steam, or a local steamclient.so.

==

The error seems particularly egregious since it is sold by GOG but has Steam dependencies.

With Steam running, the last 2 lines don't appear but the rest does.

=============

Once it works, there is no feedback in the DedicatedServer shell script, but the log gains an entry like:-

29/07/2023 23:35:48: HOST: All requests are claimed, forming match
29/07/2023 23:35:48: HOST: We have a game session - going with it
29/07/2023 23:35:48: Got a GameSession

=============

Perhaps this will help someone else who was confused like me.