UPDATE 31st March 2023: Since publication of this piece, SEGA has been in touch to clarify a few things about Endless Dungeon’s shared progression.
«Any story progress that you get with the hero that you are playing in a multiplayer game you do indeed get to keep on the main game. So, let’s say you are playing Bunker and you advance her quest — that would still be the case on your ‘main’ game and any solo or other multiplayer games after that.
«When you play with a person who is hosting, you get to keep everything you earn, except for unlocking new heroes. So to clarify — any currency earned, quests advanced, hero chips, district keys, beverages, and any progression unlocked in a co-op game you can keep and take home to your main game.