In the Gameplay section of the options, make sure online visibility is set and/or LAN or Direct connection are enabled.
After loading a save, open the in-game menu (Esc in keyboard UI, Start/Options button for controllers) and select the Connectivity Menu, and you can adjust those settings for the current session (in the controller UI, the LAN and Direct Connection settings require a left stick press to access from there).
In order to create multiple avatar characters yourself (as opposed to recruiting companions after getting into the game) you can start an instance of the game, lower the resolution and switch to windowed display mode, and maybe in the audio option set the game to mute when not in focus, then start a multiplayer game, open the Connectivity Menu, optionally disable online multiplayer (if it is enabled) and make sure LAN Connections is enabled (or that can be done in the options before starting a new game).
Next, start up to 3 more instances of the game directly from the 'EoCApp.exe' program file (doesn't matter if the first instance is started through Steam/Galaxy or the executable), select Join Online Game in the Campaign section, switch to the LAN tab, rather than Online, and join in character creation.
If you just need 2 avatar characters, that can be done in split screen co-op mode, using 2 controllers.
There is a Party Management screen accessible from the in-game menu, where you can reassign characters.
For the online visibility option, with Steam there is a 'Friends' setting in the game, in addition to offline / invite only and public (it isn't showing up for me in the GOG version, but that may be because I don't have any GOG friends in the account I have the game).
Saves are stored on the host's system. If your friend wanted to continue without you, you would need to email him the save, or provide a Dropbox or Google drive link, etc. He could then host the next session, or return an updated save.
Each save is a folder in the '..\Documents\Larian Studios\Divinity Original Sin 2\PlayerProfiles\<ProfileName>\Savegames\Story' folder.