Ostensibly they should belong to the estate of the deceased. As it is now, in the US (as I understand it) organs and body tissue is in this weird limbo of not being considered property. The original case law around this seems to exist to: 1) make it hard to go after doctors who do things you don't like with your discarded tissue (either destroying it, or giving it to the state for evidence), and 2) prevent trafficking in humans and their organs.

The second part seems to mean that only certain types of nonprofits can accept organs. From there they may or may not be able to profit off of the sale of the organ but instead the services to transport and maintain it. Then at a certain point, you've turned penises into lip filler, and then you can sell the lip filler (at a huge profit).

I replied elsewhere in the thread, but in a tldr kind of way: I think next of kin should get the organs (or the estate, as another commenter mentioned). In the case where there is no next of kin and the deceased did not express their wishes (in a will or similar), then, and only then, should the state gain custody of the body and dispose of it as the state sees fit.

I'd disagree with "as the state sees fit".

I have no problems with my organs being used to help others, either as direct spare parts for others, or for medical research.

But I'd take issue with (well, I'd be dead, so I could no longer take issue, but you catch my drift) my organs being used for just about any other type of research, since that could include research into better ways of killing one another, which I would consider a "bad thing".

Just extending that line of reasoning, what about your organs powering someone who goes on to do such research?