There's currently no way of sharing individual news stories from the Bing News app. Hopefully, this might come with a later update, but I guess this depends on copyright agreements/usage policy for the sources.
For now, all you can do is use an alternative news app, or the web pages for the news stories themselves.
while I do like Bing News for it's ability to push breaking news, I mainly use Nextgen Reader for my newsreading needs. It allows me to share news stories using various ways, be it email, sms, chat or social networks.