The WordPress.com like button is an in-house like button. It's like every other in-house member use only when logged in like button (example Facebook like button). Only logged in WordPress.com bloggers can use the WordPress.com like button on WordPress.com blogs.
Also note likes and shares are misleading when you are talking about page view stats. Your followers and anyone with a WordPress.com blog who is logged into WordPress.com can "like" and "share" your posts in several locations such as the Reader without ever clicking into your blog and creating a page view stat. See here for what I do to compel readers to read posts on my site > http://onecoolsitebloggingtips.com/2012/07/09/wordpress-followers-likes-and-stats