Clicking on the JetPack sharing buttons in WordPress opens a new window with the parameter
?share=... added to the URL. However, instead of being redirected to twitter or Facebook or wherever you wanted to go, you’re redirected to the post itself again.
In my case, this was caused by the “Redirect ugly URL’s” setting in the popular Yoast WordPress SEO plugin (marked as “not recommended”, obviously for good reasons):
After unchecking this option, sharing worked as expected.