This is a bit of an addendum for people in my situation - I had exactly the same problem, but couldn't find the "Enable simple file sharing" checkbox. After a bit of searching (which I'm hoping to save some people with this post), I found out that it's because I was using XP home edition - which doesn't let you disable simple file sharing.
There is a way around it though, detailed at this site - http://wiki.ehow.com/Disable-Simple-File-Sharing-in-Windows-Xp-Home-Edition.
After following those instructions, I still had some problems accessing the actual "My Documents" folder in the old harddrive's "Owner" folder. This is where I'm getting a bit sticky - I don't know if what I'm doing is correct, but in the properties dialogue, there is a tab labelled "owner" or "ownership" (it's easy to spot). In that, I've changed the ownership to the new computer's administrator. It seems to have worked, but I'm mildly worried that the original "owner" folder from the new computer has been overwritten.
EDIT
It's worked, everything is fine. If anyone knows that I've done something wrong, please throw a post here to warn other people away from my methods - I'm patching all this together as I go along...