In what amounts to a policy paper published in volume 7 of Techumin (1987), the Chief Rabbinate of Israel outlined their position on heart transplants. They begin by acknowledging the position of Rav Moshe Feinstein and Rav Isser Yehuda Unterman, who forbade the practice when it was still relatively new. They then note that over the decade and a half since heart transplants first came on the medical scene, the science had progressed so significantly that now roughly 70% of recipients were living at least five years. They noted as well as letter from Rav Moshe Dovid Tendler, son-in-law of Rav Moshe Feinstein, stating that Rav Moshe had changed his views on heart transplants in light of the advances made in science since the time of his original teshuva.
As such, the Chief Rabbinate was willing to allow heart transplants to take place in hospitals in Israel provided that certain criteria were met. With regard to the donor, there had to be a clear establishment of the death of the donor, as determined by confirmation of brain stem death and cessation of respiratory activity. There also had to be written consent from the donor's family as well as the presence of an agent of the Chief Rabbinate present as part of the team that determined death. Note that there is no longer a concern that the transplant would qualify as killing the recipient, as most recipients did succeed in living somewhat normal lives after the procedure.
The article concludes with a list of clinical indicators that brain stem death has, in fact, taken place.