Some places use a dedicated Twitter account
There's also the ever-popular email or text path
To an extent, however, there's no foolproof/failproof way to accomplish what you're describing: phones could be down (of folks don't see the message), Twitter could be down, statuspage.io or AWS could be down...
Also - being "down" only matters if you're trying to use whatever is down while it's down: I don't care that you had an outage last night while I was asleep ... unless I do care
If you get it fixed fast enough, no one will care (though in-depth postmortems are great to read sometimes)
