The whole energy.The photon is either absorbed,or emitted.If that didn't happen,then the vertex of QED would have at least 2 photonic lines...This is not possible,even if assuming one of the 2 photons were virtual.
If you have a bound electron (like in an atom), then the whole energy is transfered.
But for free electrons it is different. The photon can transfer only part of it to the electron.
"In the Compton effect, individual photons collide with single electrons that are free or quite loosely bound in the atoms of matter. Colliding photons transfer some of their energy and momentum to the electrons, which in turn recoil. In the instant of the collision, new photons of less energy and momentum are produced that scatter at angles the size of which depends on the amount of energy lost to the recoiling electrons."
from http://www.britannica.com/nobel/micro/138_88.html [Broken]