If somone donates part of a liver for transplant, do they

and likely 2 surgeries at the same time, donor > recipient.

needing the organ is Medicare eligible. I bet it is handled similarly by commercial insurance.

With a kidney transplant, the donor's costs are covered by the recipient's insurance/Medicare.

whether it's a living or cadaver donor.

There might be a medical bill if there are complications and the hospital stay is prolonged for any reason.