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Quake 4

From the store page:
Please note: Multiplayer is available only through a LAN connection
Heretic: Shadow of the Serpent Riders
HeXen: Beyond Heretic
HeXen: Deathkings of the Dark Citadel
HeXen II

All 4 of them mention this in their store pages:
PLEASE NOTE
Multiplayer is only available via LAN
Powerslave (DOS version)

From the store page:
Multiplayer: TCP/IP LAN
Aragami 2

"Public lobbies don't require GOG GALAXY, however it is required for setting up private lobbies with the invite option. "
Monster Train - requires Galaxy to access online features.
I love lists like this. Thanks for all the hard work.
I think information for AvP Classic 2000 needs to be updated, I could easily host/join a match yesterday (offline installer). I didn't find any active sessions, though that can be explained by joining in weird hours... Can someone else confirm just to be sure?
Another one with DRM'd multiplayer! Yay!

Roboquest

From the store page:
Multiplayer is available between GOG GALAXY users. Cross-play with Steam users will be added in one of the upcoming updates
Store page link:
https://www.gog.com/game/roboquest