Does the Code You Run Here Ever Leave Your Browser?
No -- and the sandboxed worker is built specifically so that answer stays true, not just stated.
JavaScript challenges run inside a Web Worker -- a background thread the browser itself provides. That worker has fetch, XMLHttpRequest, WebSocket, EventSource, and importScripts deliberately replaced with stubs that throw if called. That is the actual mechanism behind the claim: your code could try to make a network request, and it would fail immediately, because the capability to do so was removed before your code ever ran.
Press Run to see what your code does.
Run that and you'll see it fail instead of quietly succeeding -- that failure is the sandbox doing its job, not a bug. Pseudocode and Python challenges never had network access to begin with; their interpreters simply do not implement it. No execution service exists anywhere in this stack for any of the three languages -- there is no server this code could be sent to even if something tried, because grading and running both happen entirely on the page you're looking at right now.
Everything above runs in this page, on your own machine. There is more of it on the practice pages.