Why these run in your browser
Most of what gets pasted into a developer utility is not public. It is a production API response, an internal log line, a token from a staging environment, a config file with a hostname in it. Sending that to a stranger server to have whitespace added to it is a poor trade, and it happens constantly because the convenient tools are the ones that do it.
The formatters, decoders, generators and converters here run on JavaScript already in your browser. What you paste is parsed on your own machine and never transmitted, which is what makes them usable on the material you would otherwise have had to format by hand.
The ones that do need a server, and why
A handful genuinely cannot run locally. Catching a webhook requires a public URL for something else to POST to; sharing a snippet requires somewhere the other person can fetch it from. Those are marked as such rather than quietly grouped with the rest.
Even there, nothing asks you to register. What you create is addressed by an unguessable URL and its management token is kept in your browser, which is what lets these work without an account rather than in spite of not having one.