When a reply is made to a comment, a twitter notification (@twittername) will be sent to the original comment author to notify him/her of the reply. The notification will be sent from your Twitteraccount. 
This will allow the users to follow up on the comment and expand the conversation if desired.

CommentTweets

When a reply is made to a comment, a twitter notification (@twittername) will be sent to the original comment author to notify him/her of the reply.

nickgrossman

Enable commenting on your posts via Twitter.
Posts will get a “Discuss on Twitter” button, which will prompt visitors to reply to the original tweet about the post, as well as a “View Discussions” button which will link to existing tweets about the post.
Each new post will be auto-tweeted (requires the <a href="https://wordpress.org/plugins/wp-to-twitter/" rel="ugc">WP to Twitter</a> plugin and a <a href="https://developer.twitter.com/" rel="nofollow ugc">Twitter Developer account</a>).

CommentTweets vs Discuss on Twitter