Basically, what you need to solve in something like p2p twitter is: how to authentificate the user, how to send the message around, how to archive the message that it is not lost.
Now the genius idea: all of this is exactly what bitcoin solves.
Bitcoin already has mechanisms for authenticating. Bitcoin already has mechanisms for sharing messages (altough it is called "transactions"). Bitcoin has a mechanism for archiving those messages (it's called "blockchain").
And the kicker - bitcoin already has an ability to enter arbitrary data into the transaction. (sure, not much, but about the 160 letters tweets have.) You just add it into the script and add OP_DROP; since the transaction including script is signed, noone can just remove it.
My idea was to hijack the already existing bitcoin infrastructure and hack Twitter-like P2P messaging thing on top of it; let's call it "bitmessage".
How would sending a "bitmessage" look like? Well, you can, for example, send money from your address to the same address (send money to yourself) and add the wanted string into the script.
You can then look for a message of a given user (if you take bitcoin address as an ID of the user).
Now of course, this wouldn't be free for long (if it catched up). Right now, bitcoin miners take just anything the P2P network throws at them, but if the network was flooded with people sending money to themselves with a message in script, they would just throw these away so it doesn't enlarge the already giant blockchain. However, you can add fees to any transaction. If you add fees high enough, some block miner will eventually take your "bitmessage" and add it to the blockchain.
Would the resulting thing be equivalent of twitter? No, it would be slow (it takes about 10 minutes to get transaction to blockchain) and it would most likely not be for free (if it catches on, miners will throw away the messages with low fees). On the other hand, every message would be recorded forever, without any chance of anyone deleting it, censoring it or getting your IP address. Also, the Bitcoin seems to be going strong and doesn't seem like it will go away any time soon, and the infrastructure is super-strong.
I want to hack a prototype... maybe... one day.