It’s hard enough for most people to imagine what it takes for a living donor to give an organ to a friend, family member or stranger. What about those who have done that good deed twice?

Only 47 people in the U.S. have donated more than one organ to two different people over the past 25 years, according to the United Network for Organ Sharing (UNOS), the nonprofit that runs the nation’s transplant system. (Another 17 people have donated two different organs to the same recipient on different dates.) Of those who donated to...