No subscription is required for the base version no account, either. Of course, the base version doesn't have all the features as premium, but it is still excellent, in my opinion. I have used the free version for years. There is an offline version, as it is open source.
The full text isn't stored on the servers, but you can always just deploy it yourself for maximum privacy:
There is no reason you can't deploy the Java server nor run the docker container on Linux. Or do you mean a native client for Linux?
For my Ubuntu machine as a client, I just use the web extension for Firefox and I also use the Visual Studio Code extension. I am most happy that my READMEs don't have such stupid errors anymore when I push them publicly where anyone can see. LOL
I run the Java server fine on my Lunix box and clients on Linux & ChromeOS use the browser LibreOffice plugins to connect to it over HTTP. Works great.
The full text isn't stored on the servers, but you can always just deploy it yourself for maximum privacy:
https://dev.languagetool.org/http-server
Furthermore, there are options for docker: https://github.com/languagetool-org/languagetool#docker