I'd like to think not ALL duplicate content is bad.
For example, in your situation the other party is posting content from your site that you already published.
As long as your site has been crawled and that page is already in the SERPs, and the new website puts a link to the original article on your website, I don't think there should be any issue.
If I go around and copy/paste other people's articles on my blog, are those people going to be penalized? Highly doubtful. If anything my website will be penalized because it's full of a bunch of duplicate content I just copy/pasted word-for-word.
I personally would re-work a new article or ask them to summarize your article, to be on the extra safe side. But that's just me.