Is there a way to play through a story-driven campaign in Starcraft 2 in a multiplayer environment? If its not built-in, are there community-created multiplayer maps that have this sort of feature?
The story driven single player campaign can only be played by one person and cannot be played by more than one player.
However, there is one mission that has been altered that allows 2 players to try this mission on different difficulty settings. http://us.battle.net/sc2/en/game/maps-and-mods/mods/left-2-die
As far as community created multiplayer maps, there are numerous survival types of maps out on the custom games list that players play versus an AI, some are story driven, some are survival. Select join a custom game, then in the category drop down select "Co-op vs. AI"
Edit
As of the latest expansion there are coop missions that can be completed with friends. The missions range in difficulty and characters that you can choose. They are designed to be completed with a friend. It is called "Allied Commander" mode. There is a much longer blog post from the Blizzard website, and you can check out the preview video here.
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1it may be worth updating this answer as the new expansion legacy of the void has added a co-op campaign mode (but only for that expansion) – jammypeach Nov 12 '15 at 13:56
