Some word of warning here:
Event flag progress is saved on the Name Lobby.
Your item progress is saved on your actual save file.
You can technically join someone's lobby with a completed save file, and you can inject all your items into the player's lobby, potentially ruining the game.
Make sure you work with the host before you join a game to avoid causing issues.
Otherwise, if you need to save and quit due to time, or bugs, save the game, and rejoin the lobby with the save game you were using to continue on from where you were at.
Also, enemies and time cycles are not synced. It's very possible to be out in the field during the day while you are watching someone fighting stalphos at night, and you can't help them.
Boss HP is synced, but not their moves. You all fight your own boss, so you'll watch people on your screen slashing at air.