there are basically two possible ways how to deal with it.
1) Everyone is presumed donor, unless he/she states otherwise. This requires some sort of nationwide database of non-donors or perhaps having such thing mentioned on your ID
2) Everyone is presumed non-donor and permission is sought ex-post. This might lead to many organs being wasted since many of them are "useful" only for a quite short period of time, I believe it's something like 4 hours in case of heart. Also dealing with relatives in such a traumatic situation when they just lost a loved one is difficult for both parties.
Should situation like this ever happen to me, I'd be more than happy to be used for "spare parts".