I have been asked the question posed in the title of the post many times, and each and every time I answered it on this site, I was emphatic that this could not be done because of the problems that we see in kidneys, livers, and hearts when they are transplanted (i.e. they get rejected by the body). Now, the possibility that this may not be the case was raised by Dr. Sharon Keene after she read this article in the LA Times about a small pilot study which may lead to the elimination of taking anti-rejection medication for life following organ transplants — Study suggests breakthrough in organ transplants.
Japanese doctors tried homografts and allografts of scalp hair back in 1928 and again in 1938. They left detailed descriptions of the results. The growth rate was zero percent; however, we have learned much about transplanting organs since then and maybe we can apply what we learned to allow for a better outcome.
Of course, I am not saying that my answer has changed, but who knows for sure what could happen. Successful transplants have been done between different people, in baby hearts under a year of age and in face transplants where the anti-rejection medications can be far less and possibly stopped, but the experience is still very sketchy, as not many face transplants have been done to really know what it takes to sustain such a transplant without anti-rejection drugs.
There are also ethical considerations and medical-legal ones imposed on doctors who think that they could experiment on people without legal ramifications. When I was in medical school our most famous surgeon was Dr. David Hume, who made his reputation in Boston at the Brigham hospital doing a kidney transplant from one twin to another in the wee hours of the night without the permission or knowledge of the medical staff. It worked and he became a hero, but had it not worked, he would have become vilified and this act could have ended his career. With stem cell progress and some of the tricks outlined below, who knows what the future holds; however, I have long believed that the really important problems we, as doctors, will be allowed to treat, will not initially be for the vanity of perfectly healthy men who are balding.