No server-side processing - and your data never leaves your pc.
If you host it locally, you can even turn off your internet connection after the initial download.
You're also able to upload and chat about files (pdf, txt, code files and more).
Check out all the features in the open-source github repository. You're also more than welcome to contribute!
Github repo: https://github.com/addyosmani/chatty
Deployed app: https://chattyui.com/