(Sorry in advance for this russian novel length posting. This is just a subject I am fascinated with.)
I have actually done a lot of research into organ recovery and transplantation. I wrote a story about it last year. I have to tell you, after everything I read, I still could not make up my mind.
One thing that really perplexes me about organ donating is the fact that most donors are what is called a "beating heart cadaver." What that means is that the person has been declared brain dead and has no brain activity, which is the medical definition of death. But, due to the miracles of modern medical technology, the person's body is still "alive." By using a respirator and careful controlling hormones going into the body, doctors are able to keep the person's heart beating and their vital stats in check. It's amazing. They can do this for several days if they need to. Then, when it is time to recover (they used to call it "harvesting," but that was too crude) the organs, they pull the breathing vent and start to cut open the donor while their heart is still beating. In fact, they will actually remove the heart while it's still beating, and hook it up to special fluids which can keep it beating on its own inside a container for hours).
What keeps me troubled about this (and this is just a personal opinion. I am not trying to dissuade anyone or inject an opinion here) is that even though the donor is brain dead and will never return to a state of aliveness, their body is still very much "alive" and being kept in a viable state so that the organs will be able to survive transplantation. Part of me just has to wonder if because of that, some part of the person's lifeforce is still there, in the body, and that the soul is experiencing some type of agony because it is not fully "dead" yet.
All of my ideas are just based on my own fears and ignorance of life. The science of death and lifeforce is very limited, but all medical professionals agree that when a person is brain dead, they do not experience pain or anything. I just can't be so sure.
For me it's not a matter of science, but a matter of faith. What do I really believe happens to our body and soul when we die? I don't know. I know what I would like to believe, but the only way we ever know is when we die. You have to have faith that what you believe is correct. I think I would leave the choice of donating my organs up to others, really. It's a decision I can't make.
There's a great book that talks about the process of organ transplantation called Stiff: The Secret Life of Human Cadavers by Mary Roach. It's not for the squeamish, but I highly recommend it if you're fascinated by this stuff, which I am.