There are two main systems for deciding what happens to someone's organs when he dies. The first is the opt-in system: unless an individual registers to be an organ donor, he is presumed not to be one. The second is the opt-out system: unless an individual registers not to be an organ donor, he is presumed to be one. The opt-in system is also known as "explicit consent", while the opt-out system is also known as "presumed consent". It is well known that the effective donation rate is considerably higher in countries with the opt-out system. In other words, many more organs are available for transplantation in countries where people must actively opt out of being an organ donor.
The opt-out system is quite obviously superior to the opt-in system. You get a higher effective donation rate, while still giving people who do not wish to donate their organs the option of not doing so. However, there is a problem with the opt-out system: it permits free-riding. An individual who opts out of being an organ donor is still entitled to receive somebody else's organs, which means she has too weak an incentive not to indulge her (inexplicable) preference against being an organ donor. (I use the term 'inexplicable' because I personally find it very strange that somebody should care what happens to her organs after her death.)
Luckily, this problem could be solved with a relatively minor tweak. Just like in the conventional opt-out system, every individual would be be an organ donor unless he had registered not to be one. But in addition, no individual's organs would go to a non-donor unless he had registered to allow them to do so. Therefore, the default situation for a newborn baby would be as follows: it is presumed that he is an organ donor, and it is presumed that his organs will not go to a non-donor. Under this alternative system, there would be a cost to opting out of being an organ donor, namely that most people might not be willing to let their organs go to non-donors.
If a person felt strongly that anybody should be entitled to her organs––even people who had not been willing to donate theirs––she could register to allow her organs to go to non-donors. But if a person felt that only people who had been willing to donate their organs should be entitled to hers, she would not have to do anything. My guess is that the effective donation rate would be even higher under this alternative system than it is under the conventional opt-out system.