Following document says only the owner of the channel can do this via standard youtube interface: https://developers.google.com/youtube/2.0/developers_guide_protocol_captions?hl=en
Cheap fix: You can click on the "interactive transscript" button - and copy the content this way. Of course you lose the milliseconds this way.
Extremely cheap fix: A shared youtube account - so that multiple people can edit and upload caption files.
Challenging solution: The youtube API allows downloading and uploading of caption files via HTTP... You may write a youtube API application to provide a browser user interface for uploading or downloading for ANY user or particular users.
Here is an example project for this in java http://apiblog.youtube.com/2011/01/youtube-captions-uploader-web-app.html
Here is very simple example of a working upload for everybody: http://yt-captions-uploader.appspot.com/