Major mechanisms for depositing the energy is through Compton scattering and/or pair production -- the gamma photon "hits" an electron (or produces a pair), and ejects it, and then these electrons interact with the other atoms in the material and so on.
So the steps are:
get the photon to eject one or more electrons,
the electrons interact with the material via bremsstrahlung, which is more
effective with higher density materials (and higher atomic numbers), to give off X-rays,
- Go back to (1) (i.e. the bremsstrahlung X/gamma-rays ionize other electrons)
Repeat until you get down to non-ionizing radiation.
Due to the cross section of the photon scattering processes you need a lot of depth to ensure that almost all of the photons and electrons in the cascade interact with, and stop in, the material. High density materials provide more opportunities, per unit distance, of having one of these discrete interactions. Also note that alot of the Compton scattering is off of electrons in the inner shells of the atoms, and do not involve the conduction band electrons. All of this is well described in terms of particle "ballistics" and does not involve inducing a normal, bulk, electric
current in the material (which would be influenced by the conductivity).
Although classical evaluation of the skin-depth indicates that higher conductivity would be more effective, it ignores the quantum-mechanical effects that are relevant because a single gamma ray photon can have so much energy (relative to any of the energy scales that characterize the material).