What happens to followers when playing multiplayer? Can they be used?
For both followers and non-follower NPCs, any quest-reliant companions will continue with you on any quests in which they are relevent - initially, this includes the Templar, the Scoundrel, and the Enchantress (all for different quests). At that point in time, regardless of whether or not you are playing multiplayer, they are not actually considered "a follower" - you cannot adjust their abilities or inventory.
After they have been recruited as your follower, it will prompt you as to whether or not you wish to keep them. If not, you can hire them again in town any time you are playing alone in a game. If you are playing with multiple players, followers cannot be hired. If you have a follower when you game switches from single player to multiplayer (i.e. someone joins your game or you join someone else's game), your follower will be "un-hired" (teleported back to town, where you can re-hire them next time you are alone again).
As mentioned by @Hurda in the comments, when all co-op players leave, the game will prompt you with the option to re-hire your previous companion, who will join you again without having to go back to town.
TL;DR You cannot hire a follower in multiplayer, and anytime you either gain any or lose all co-op players, the game auto-dismisses or prompts for re-hire.
In multiplayer, followers will stay with you during the quests for which they are required. For example, the templar will follow you unitl Jondar is slain, and the enchantress stays with you until she reveals the bridge.
Once you would have the option of whether to keep them or not, they go back to town automatically. You can not choose to use them in a co-op game.