Messages aren't showing

What do you mean by “message feedback”?

Things like: “Your message was successfully sent” etc.

Can you replicate the problem using the default Twenty Fourteen theme with all other plugins deactivated?

No, it wasn’t replicated in the standard theme, but still a plugin issue.

I went through the js code, I noticed it calls
$form.find("div.wpcf7-response-output");

However, for some reason, the shortcode has output the response outside of the form. The output is showing the structure:

<div class="wpcf7">
  <form></form>
</div>
<div class="formItem">
  <div class="div.wpcf7-response-output">
  </div>
</div>

Ah, there was a stray </div> hiding in the template. No bug to report!