Would the brain be damaged, or would everything just stick itself together instantly?
Adding to the above answer, the most immediate fatal events are:
While you can cut the corpus callosum effectively separating R/L cerebral cortex without killing the person, splitting your midbrain/brainstem is instant game over. Almost all signals to/from the rest of the body and cerebral cortex, including your heartbeat breathing and consciousness, originate or go through there. There are multiple points of crossover that you'd interrupt - there and throughout the spinal cord.
You'd transect critical vessels (the aorta at the arch, superior/inferior vena cavae, circle of Willis, dural sinus) so you'd both dump out massive blood volume in an instant, and stop all further circulation. Vessels don't auto-seal, even a small break can cause significant leaks. Transecting your CSF ventricles would drain the fluid from the brain.
A person is not a loaf of bread, but a network of organs at various positions that depend upon each other. What you are basically asking, would damage some organs by cutting them in half this rendering them useless, while others you would only cut off the blood supply, which would kill them. TLDR they would bleed out before anything else could happen.