Each website is only able to access and manipulate its own cookie, but you can use frames and iframes to display a different website's widget on your site (and allow it to access its own cookie).
In the case of Facebook's button, you're referencing a PHP script and Javascript on Facebook's site, which are able to access the Facebook cookie.
For example:
<a name="fb_share" type="button" share_url="http://example.com" href="http://www.facebook.com/sharer.php">Share</a><script src="http://static.ak.fbcdn.net/connect.php/js/FB.Share" type="text/javascript"></script> If you want to restrict a site's ability to access its own cookie, you would have to either configure your browser settings as you usually would (some browsers can be configured to always prompt you whether or not to allow cookies for certain sites), configure something like NoScript to prevent the JavaScript from running unless you explicitly allow it, or set up a firewall rule that blocks traffic to the site in question.