when a user logs out, the cookies on the client machine are deleted, or expired.
whenever a cookie is accesed you can check the expire date on it.
but, yes if the user is not logged out or the session is not destroyed, then i think you can copy the cookie over and try it. i have never tried this, but sounds it might work, unless we have some kind of machine_number registered on the cookie, so it only works from the computer it was created on.
this is the reason, i myself dont use cookies or sessions till now. they are not very, or shall i say 100% reliable if you are dealing with sensitive data.