I'm broadcasting this as a helpful tip to anyone else who's as oblivious as I am, but after many broken saves that we can't load, and strange hangs on 'Loading Game Data' screens (or whatever, after the host had loaded and was waiting for his computer to communicate with mine--Task Manager informed me that we weren't actually transmitting data, though.

Symptoms included:
Being unable to join his games in a multiplayer lobby (but being invitable, nevertheless)--The game would indicate I could see his game, but I couldn't join because our files were different.

The game would start fine, and randomly crash. The first couple of crashes, I could be invited back into the game again.

Mods behaving strangely--The world generation mod in particular decided it did *not* want to distribute luxuries--At all!

The solution? Matching the mods we were using and ensuring they were loading in the same order!

spam:
Forgive the following phrases--I want to help others who searched as I did find this solution. Endless Legend Multiplayer stuck on game load, Endless Legend multiplayer stuck on loading game data, Endless Legend multiplayer cannot load save

/spam

Tl;dr: Broken save games? Weird multiplayer crashes? Using mods? Synchronize your mod load order!