The Tweet Button and Web Intents don’t directly reveal any personally identifying pieces of information when an end-user makes use of them. With Web Intents events, you can detect the fact that a tweet occurred, but not the specific tweet id or user who did the tweeting.
However, with the streaming API you could continuously monitor tweets with URLs pointing to your domain name (if that makes sense for you), which would give you some insight into the users who are tweeting about your site.
