Jetpack Comments and the other Jetpack modules will only work once you link Jetpack to your WordPress.com account. This allows you to use some of the WordPress.com resources on your site.
You don't have to use all the Jetpack modules though; if you want to use Stats, but not Jetpack Comments, you can follow the instructions here to learn how to disable only the Comments module:
http://jetpack.me/support/#turn-off-jetpack-modules
And when I comment on anyone using WordPress.com it won't let me comment using my wordpress.org account just my .com one.
Your WordPress.org account credentials will only work when trying to log in to your own site. No other site has any information about your personal site credentials; that's why you can't log in to WordPress.com using these personal site credentials.
I hope this helps.