It needs to be Open Source so anyone can run it and everyone owns their data.
It needs to be compatible with current Twitter apps so all it requires is setting the API root.
It needs to be distributed so anyone can follow anyone anywhere. There is no owner or root, there is no place to shut down.
Proof of concept: https://nstatus.herokuapp.com
Source: https://github.com/maxpow4h/nekomimi
I wrote about the requirements of it here: http://maxpow4h.com/blog/twitter/
edit: you can use any username with any password to sign in to nstatus. It then uses that password for your username. You can even do this from the official iOS Twitter app, just sign in.