That might be possible to implement, actually. Just like how clicking on facebook spotify 'listen with' links now plays the song in the _spotify_ app, there may be a way on the server to control what the client does...
A custom protocol wouldn't strictly be necessary. You could have the browser tell the server the path of the dropped files then have the server instruct the native client to fetch and upload those files.
However the File object hides the full path from the JavaScript code for security reasons (see https://developer.mozilla.org/en/DOM/File). I assume most dropped files would contain the user's home directory and therefore account name in the path.
The client has a permanent connection to the server, but in this case the server doesn't necessarily know what client it's on, and some of us have our browsers configured to be pesky (with a dialog box indicating the URL and target app) about sending URLs to different apps.
On Windows at least, this requires a registry entry to associate the application with the new "protocol". Steam does this as well to launch games and so forth with "steam://foo"
However the File object hides the full path from the JavaScript code for security reasons (see https://developer.mozilla.org/en/DOM/File). I assume most dropped files would contain the user's home directory and therefore account name in the path.