In chapter five of the manga, an ajin remarks that:
An ajin uses the largest piece of flesh as a base and collects the
smaller pieces for regeneration. However, pieces that are too far away
are not collected, but replaced instead.
Later in the series (I forget the context), we see an instance of an ajin's head being severed but it was far enough away from the rest of the body that the regenerating body simply discarded the old head and grew a new one, but the old one appeared to remain intact (presumably just decomposing at the rate a human head would).
In the anime, I assumed that Sato was having his own organs removed, letting himself die and - if the extracted organs were moved far enough away - his regenerative powers would simply replace them. By the time we see Sato making dealings for spare organs, he has a few ajin following his crusade and so they too are probably just volunteering to have their organs harvested and replaced.