I've often wondered the same thing. For me, I saw Andrew's question as an opportunity to set up an instance of GNU social, which is an open source, federated social network from the Free Software Foundation with a number of Twitter-like features. What does it mean to be federated? It means there can be many GNU social servers and a user on one server can communicate with users on other servers.Dear #MTBoS,— Andrew Stadel (@mr_stadel) October 8, 2016
If Twitter bites the dust, what's your contingency plan for maintaining your Professional Learning Network?
We've grown accustomed to companies running social services designed to keep us within their site. GNU social is more like email: it doesn't matter if you use Gmail, Yahoo, your work email, or you set up your own server in your basement. Every email server can talk to every other email server. If the math ed community ever wanted to go rogue and leave Twitter behind (or Twitter left us), I can foresee the possibility of a social.mathed.net, a social.globalmathdepartment.org, a social.desmos.com, etc., and we'd be free to establish an identity on whichever one we chose and we could still follow and chat with those on other servers.
Now that you know a little about the technology, I'm inviting you to join a temporary "social sandbox" I've set up just so I can learn first-hand what running a social site entails. The installation wasn't difficult -- it probably took me less than an hour to install and configure. It's running on a little test server that might be easily overwhelmed, but that's okay. Just consider anything you post to be disposable, as I will take down the site before the end of 2016.
Try it at socialsandbox.mathed.net.