Page numbers aren't part of a PDF's structure the way the page count is — they're just text, drawn onto each page like any other content, at a position the original author chose or didn't choose at all. Adding them here means opening each page, picking a spot near an edge, and drawing the number in a plain system font sized to stay out of the way of whatever's already there.
Because the text is drawn directly onto the page rather than added as a special PDF feature, the result opens identically in any PDF reader — the numbers are just part of the page from that point on.
Everything runs in your browser. Your file is never uploaded, never queued on a server, and never deleted from one — because it was never there.