Up, I'd like the info too.
If my colonist is rejecting a new kidney, can I try to put other kidney and see if he rejects them?

There is a way to remove this "mild rejection 0.1%"-no-matter-how-often-you-treat-it-it-won't-go-way state but it requires some config modification.

In the xml config files in steam\steamapps\workshop\content\294100\850429707\Defs\HediffDefs open the config you want to edit ("HediffDefs_Natural.xml" for example is the config for natural organs). you can manually add a new "rejection state" called "no rejection" or whatever you want to call it. Just search the file (ctrl + F in most text editors) for <minSeverity> and set it from I believe 0.001 to 0. Then search for <stages> and add
<li>
<label>no rejection</label>
<partEfficiencyOffset>-0</partEfficiencyOffset>
</li>
as the first element.
I'm not too familiar with modding in rimworld so I don't know if you really have to set the partEfficiencyOffset to -0 but I did it to ensure it is properly set and overrides previous penalties.
Finally you have to set <minSeverity>0.01</minSeverity> in the "mild rejection" section. Then restart the game. Once ingame it will show for example:

Left kidney Kidney transplant (no rejection) (and that white circle --> don't know how to remove that)

However all part efficiency penalties are gone and no more treatment is needed.
(Btw. it is compatible with save games)

The rejection on natural organs will last forever, like in the real world, but the symptoms are mild when managed. You retain 95% function. In the game you can replace them with synthetic organs, which do not have rejection symptoms.

The only reason I wish the rejection would eventually go away is the Body Purist trait. I guess I will have to use synthetic organs for that? That seems like a waste.

This was going to be a very important question for me, as I was finally going to do a heart transplant. And then I discovered I needed a stomach, not a heart. *facepalm*

Do you need more than Anatomy/Physiology to do a transplant?