The constitution doesn't apply to a location or a medium, it applies to an actor: the US government (and state/local subdivisions). The US government has to follow it everywhere, and nobody else has to follow it anywhere.
Americans do tend to run around quoting their first amendment rights like the whole world has them. As you say, it's strictly a US government thing.
While Australia for the most part enjoys free speech, it is not enshrined in the (AU) constitution. The government will occasionally order censorship[1], usually around whistle-blowing, investigations and court cases.
There was a protest in the major Australian papers last year about the erosion of press freedoms[2].
[1]:https://www.nytimes.com/2018/12/14/world/australia/australia... [2]: https://www.bbc.com/news/world-australia-50119559