Here's my theory on this whole Heartless debate:
Heartless are creatures born from the darkness is someone's heart. Knowing this, you would know that they do not have hearts themselves. Rather, they collect hearts to replace the empty feeling, but the space is never filled. Now, the people who are saying that they see hearts when they kill Heartless are, indeed, correct; you are seeing all of the hearts they have collected merged into one heart.
The Nobodies are formed when someone loses their heart to the Heartless. They are the empty shell left behind. This explains why Organization XII wants Sora to kill the Heartless. It would leave more empty shells that become Nobodies.
Now, about Kairi having a Nobody:
Kairi's heart was with Sora. Well, in order to keep his heart from being taken, he turned himself into a Heartless, therefore creating Roxas. Even though Kairi had regained her heart, part of it was still with Sora when he became a Heartless. That was when Kairi's Nobody was created.
When I told her I loved her, she just sat in silence. I knew then that love would never happen for a girl like me.