I'm probably about to make an idiot of myself, but....
I thought it went like this:
Close, personal, special memories are tied to the heart. All other memories that aren't as important or special are not tied to the heart.
For example, being in love would probably count as a close, personal, special memory, so those specific memories of being in love would be tied to the heart. Where a person was born or where they lived would probably not be a close, personal, special memory, so the memories of where that person lived would not be tied to the heart. So if this person was turned into one of the "special" (for lack of a better word) Nobodies, they would remember where they came from, but they wouldn't remember anything at all about the person they were in love with.
So the "special" Organization XIII Nobodies do have memory, but they can only remember the memories that were not in their hearts.
I am probably wrong about this, I don't know, but I thought that Heartless were not hearts themselves, but the embodiment of the darkness in a heart. That's why Sora was turned into such a weak Heartless: he didn't have that much darkness in his heart. It's also why Kairi was not turned into a Heartless at all: She didn't have any darkness at all in her heart.
The embodiment of darkness in a heart would not hold close, personal, special memories; the heart itself would. If Heartless are not hearts themselves, they would have no memories and would act purely on instinct. So Heartless would not have their memories.
But I'm not sure. I am probably wrong. This is just what I thought was actually in the game. (Is it...? If not, then what is actually in the game? Or was it even said in the game?)
Based on what I thought, I would say (to your question) that memories are tied to both the heart and the soul. Just that more important ones are tied to the heart.
Just thought I'd throw in what I thought....
Oh wow, big wall of text. I'm sorry. I probably didn't help at all, either. I sound like an idiot...