Sort-of, but the left is typically used because it is typically larger and has a longer renal vein. See Left versus right deceased donor renal allograft outcome.
Also interesting, the original kidney is usually left connected unless it’s problematic:
Once you’re asleep, your doctor makes an incision in your abdomen and places the donor kidney inside. They then connect the arteries and veins from the kidney to your arteries and veins. This will cause blood to start flowing through the new kidney.
Your doctor will also attach the new kidney’s ureter to your bladder so that you’re able to urinate normally. The ureter is the tube that connects your kidney to your bladder.
Your doctor will leave your original kidneys in your body unless they’re causing problems, such as high blood pressure or infection.
See What is a kidney transplant?.
That implies that a misfunctioning kidney isn’t necessarily a sign of deterioration that will pose an additional hazard to your body over time.
What? How? Wonder. 