No. The file is read by the browser's FileReader API and encoded locally — it is never sent over the network. That is the main reason to use an in-browser converter for anything that is not already public.
Should I inline images as Base64?
Only small ones. Base64 inflates a file by roughly 33%, and inlined data cannot be cached separately, so a large inlined image is re-downloaded with every HTML or CSS change. The usual guidance is to inline below about 2–4 KB — icons and tiny placeholders — and serve everything larger as a normal file.
Which formats work?
Any image your browser can read: PNG, JPEG, WebP, AVIF, GIF and SVG. SVG is a special case worth knowing — because it is text, it usually compresses better as URL-encoded SVG than as Base64.