I have no clue what my local government or any such authority could possibly tweet that would be relevant to me. I have not checked their twitter feeds ever and I am none the worse for wear.
Usually just information about events, road closures etc.
There is no way twitter is the sole channel of this over say pushing a text message like they do every other week with a silver alert. Just ridiculous to imagine twitter is some disaster tool.
There is a state government website with some of the same information, but it's usually more coarse (sometimes just the open/close status) and updated less frequently.
Compare that to email, which you’d get hundreds of messages a day and only care about 3 of them, so you’re not constantly checking.
Edit: dude you live in Australia. Do you know how people in the Bush get their emergency updates that they need to respond to urgently? I bet at least some of them use Twitter or something similar to it. They're not huddled around their weather radio all day (and that doesn't cover everything). Because I had family in rural California and that is exactly how they got theirs. Twitter and Facebook.