As far as I know, the Facebook send dialog does not have any callback parameters (like the feed dialog). But, I would like to be able to store the recipients of that message in
You can subscribe a call-back function via javascript on your page that has the send dialog See: http://developers.facebook.com/docs/reference/plugins/send/ and http://developers.facebook.com/docs/reference/javascript/FB.Event.subscribe/
message.send - fired when the user sends a message using the send button. The response object passed into the callback function contains the URL which was sent:
So according to the documentation you will only get the URL shared, so if you had multiple Send buttons on a page, you can tell which one was clicked.
For security and privacy reasons you cannot get the recipients of that message.