Each follicular unit of hair has blood vessels, glands, nerves, skin, and fat. It takes all this plus a growth center and the right genes and nutrition to make hair grow. Since by definition hair is an organ, transplanting the hair from someone else is like transplanting a heart, kidney, lung, etc.
One may tend to think that a person (a Family, friend or well wisher) with same blood group if willing to donate hair to another person should be an easy solution for people with hair loss but it is not that easy a solution!
Generally hair transplantation from one person to another (called Allogenic or Homologous Transplant) who could be living person or a dead person (Cadaveric Transplant) carries the similar risks as organ (liver, heart, kidney) transplants.
Transplant from the animal fur to human head is called Heterogenous or Xeno Transplant. There is a
1. Risk of rejection and failure
2. Wastes money and effort of surgery
3. Lifelong use of anti-rejection medication that could lead to further health complications
4. Transmission of some diseases.
5.It involves surgery on the donor as well
6. There are medico legal aspects to it besides ethical issues: such as possibility of people selling their hair for money in future.
Such experiment has been done before without success (http://www.hairsite.com/hair-loss-articles/article308-person-to-person-hair-transplant.htm)
Hair from a newborn baby or identical twins (they have same Genetic Blue print) might be immune privileged (it seems to work for a heart transplant), but again, who would endorse such a donor? Incidence of transplant from one identical twin to another has worked in the past but the problem is that when one twin has balding, so does the other.
There are some research indicating that Anagen Bulb part of Human hair follicle is Immune Privileged (i.e. protected from Host Rejection)
http://www.nature.com/jidsp/journal/v8/n2/full/5640115a.html ). More research may open a new vista in this field in future.
Rosati et al. reported the case that the bone marrow transplant patients could undergo an allotransplant of hair from the same bone marrow donor. Jahoda et al. (http://www.nature.com/nature/journal/v402/n6757/full/402033a0.html)