If you never extended permission to your family member to edit the tree when you first shared it, they will only be able to view, and not change, it. However, once you have given them this permisson they can access and edit whenever they want, I believe regardless of whether you change the password.
However, you are correct in saying that they will only be able to edit information pertaining to them: their own immediate family, and (I presume) relations you have in common.