For something like this, how do we know that the files are not sent to a server? Am I just trusting the web app? Is there any way to be sure other than having and reading the source?
For someone who knows that these tools exists, yes this is a way.
For an ordinary user, the only 'easy' way a user can verify the claimed behaviour is to literally go offline.
Browsers do not currently have badge to verify that the app is not sending any data. I'm thinking how we were brought to trust the padlock icon browsers display for TLS supporting sites.
I just did this and it worked.