Using BitTorrent to share (not just pirate) makes a great deal of sense. I worked on a project at the National Institutes of Health years ago to share the "open source" human genome database.
Being able to stream video more efficiently, drag-n-drop files, and other innovations could come about. One issue remains -- there is no Pirate Bay for legit torrents. By design, there is no single BitTorrent network; the protocol has no provision for search. Generally, an organization wishing to offer a file via BitTorrent would place the .torrent files on a website in the place of the usual HTTP download link. Searching for .torrent files can be accomplished through web search engines, but this is external to the protocol -- trading of copyrighted (pirated) material is not inherent in the technology.