For the fun of it, we made the protocol on top of TCP sockets, instead of JSON over websocket pull, or other alternative mechanisms. Here is a demo that shows a ChromeApp and an iOS Cordova client interacting with a couple of servers that host the tweets.
Any user can host and control their own servers, and can use any mechanism to authenticate to the server and post tweets to it. For the demo, we used a simple RESTful API, but the real power of such a system would be the ability to do things like scheduling tweets, summarizing or aggregating tweets from other followers, etc. You don't even need an account to follow others!
The client is a ChromeApp with the UI using Ionic and Chrome Sockets. Being a Cordova commiter, I also converted it to a Cordova app using mobile-chrome-app that comes packaged with appropriate plugins to fill in for the Chrome API.
The source is on github, and if you like the idea, please vote for us :)