Some journals already count tweets, and blog mentions (generally for PR reasons) but typically don’t allow access to finding them on the web to see if they indicate positive or negative sentiment or to further the scientific conversation.
I’ve also run into cases in which scientific journals who are “moderating” comments, won’t approve reasoned thought, but will simultaneously allow (pre-approved?) accounts to flame every comment that is approved [example on Sciencemag.org: http://boffosocko.com/2016/04/29/some-thoughts-on-academic-publishing/ — see also comments there], so having the original comment live elsewhere may be useful and/or necessary depending on whether the publisher is a good or bad actor, or potentially just lazy.
I’ve also seen people use commenting layers like hypothes.is or genius.com to add commentary directly on journals, but these layers are often hidden to most. The community certainly needs a more robust commenting interface. I would hope that a decentralized version using web standards like Webmentions might be a worthwhile and robust solution.