All that is passed to the social media platforms is the URL of the page being shared as a URL parameter. The social media platforms then perform a crawl of the page in question to get the details of the page to attach to the share post. Your sites cookies and referrers are not passed along as a popup window is opened which then is redirected from the addthis ...
I know it's a very old question, but you can debug that kind of problems with Facebook object debugger.
Here you can see your page OG Meta tags are messed up, I don't know why you're serving different content for Facebook (maybe all the bots?) but the culprit is the way your headlings.
[custom_headline type=" center level="h2" looks_like="h3" accent="true" ...
Your problem is that CSS with <link> should be placed within <head> </head> and not within the tags <body> </body>. This should fix your validation problems.
<title>Your Page Title</title>
<link media="all" rel="...
The page http://www.addthis.com/social-plugins/facebook-like-button states a different namespace to that mentioned above. It should be:
I've just tried it and it works fine, my counter is back.