Access your list of contacts with the "People" icon on your home screen. Select the contact for which you want to set the ringtone. If you look in the "Information" section (might have to scroll if a lot of contact information has been added) you'll see "Ringtone." Press the arrow next to that, and you'll get a list of your ringtones. Select the one you want, click "OK," and you're done! (This can't be changed by selecting "Edit.")
If the above post doesn't help, this may. I have had difficulty assigning downloaded ringtones to specific contacts as well. I found that even after downloading and assigning to a contact it was not on the list. What I did to fix it was download as the main ringtone and then it showed up on my list.