1. User requests page
2. Server generates page
3. Server sends page to user
4. Browser renders page

At this stage (although I'm simplifying a little), the page exists only in the browser's cache - and there is no way for you to retrieve that from the user's browser (short of playing around with JavaScript to post the entire page back to the server - but that would be even sillier then generating the data again (not to mention unreliable).

You might want to consider doing with the vast majority of websites do and allow the user to just copy and past the URL from their address bar into their email client. Saying "Please trust us with your friend's email address" can leard to concerns that you are harvesting addresses to spam.