Actually, P2P communication inside any browser has been possible since the release of Adobe Flash Player 10. Two Flash 10 clients can directly communicate with each other over most NATs and firewalls (in contrast to Opera, Flash 10 supports NAT punching).
Just over two weeks ago I've launched a web-based service that uses these capabilities of Flash 10. The service enables users to send files directly between each other. No software to download. If interested, you can check it at http://www.FilesOverMiles.com