The compiler is open-source and can be run locally. You need an account if you want to use their web editor, which is nice (it shows error messages where they occur along with an explanation and link to docs, and also shows a real-time updated preview).

As for Latex vs Typst, as a language Typst is much better, compiles very quickly, and has sane error messages. However, Typst still has a few rough edges, and can't do everything you can with Latex + packages (yet).

I've been using Typst for most of my documents for a few months and I've been generally happy with it.