I don't know the theme and plugin. Generally plugin print something somewhere and you need to tell it where to do that. Unless the theme a it built in calls to do so like for example rating or comments. That is completely independent from any javascript calls in the header.
So in short you probably have to modify the theme.
Forgot somehow that zp-masonry is the same developer. Sorry, not familiar.
Another idea: Make sure you don't have add blockers or Ghostery enabled as that might block addthis as it tracks of course visitors even without clicking.
Or look for JS conflicts in the browser logs
The link to the addThis plugin:
http://oswebcreations.com/zenphoto-plugins/zpaddthis/
The link to the zpmasonry theme.
http://oswebcreations.com/zenphoto-themes/zpmasonry/