If you try and define twitter and distinguish it from other types of communication, one thing that has always fascinated me that tweets could be seen as snippets from someone's thoughts. Sort of like James Joyce's Finnegan's Wake where you can read someone's thoughts. In twitter, you can act as a voyeur (as a casual follower) and just read what the others are saying / thinking: or you can also communicate with them and join in / effect their thought process.
The opportunity exists for a new type of fiction based on tweets. If you look at current attempts to use twitter to create fiction, the usual attempt is to create a work of fiction based on the individual tweets of interested participants . In my opinion, this really doesn't capture the true essence of twitter.
I am imagining a another type of fiction based on twitter where you jump into a story by reading the tweets that characters are writing - real time. Picture a story of a political campaign in which the two candidates create tweets that express their thoughts before certain events - a television debate or an election. Followers could just read and enjoy or they could contribute to the story with tweets of their own reflecting their role as campaign manager or television moderator or voter who is getting ready to cast his vote. Or another story might concern a murder mystery where the murderer and the detective each reflect on the "hunt" and "chase". Users could take over characters such as a television reporter covering story, etc. Of course, you might have sort of a War of the worlds effect where uninformed followers might think these were real thoughts. If you had too many people tweeting on the story line (maybe using #hashtags to coordinate things. Of course, the created story may get out of control if too may people start getting involved but hey life is complicated / chaotic as well.