Here's an interesting scenario: the Opera installer asks you if you want to install the "Freedom" service, if you agree, it installs a background server (always on, launched on start up). This service is a local web server + web application that provides a user interface to all your local data : your address book, your music (you can listen to your mp3s with a flash player, streamed), your calendars, your documents, etc. Of course the access is protected by default.
The networking issues aren't that problematic : the local app would periodically update a central database with the computer's IP (dyndns.org has been doing this for years) and an available open port. So when you identify yourself on the http://freedom.opera.com service, you are redirected to http://YOUR-COMPUTER.freedom.opera.com:OPEN-PORT/.
So basically, Opera would be going against the current trend of putting all your data in "the cloud". Sounds interesting to me…
The "hard parts" would be reliable dns resolution, finding common ISP open ports, and security of all kinds.