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That's going to happen regardless of there being a comments section right on the page.

Sure they can. Author could specify a unique #hashtag for the post that can be used on Twitter/FB. Not sure about other platforms that support hashtags or similar concepts, but its not out of the question to track comments like that elsewhere.

In most cases on twitter/facebook, the author's official account post is going to be the main area for discussion. For places like here or reddit, alerts are an option.