Looking most specifically regarding the moons around Saturn, Jupiter, and Uranus. I've been unable to get a concrete answer on this through various sources. I've actually read several articles briefly mentioning the exact opposites - some say the side facing the planet receives the most radiation, while the others claim it's the side facing away from the planet. Anyone have an answer?
The Planet Side.
The gas giants emit radiation. A tidally locked moon would be the only side receiving that emitted radiation. Both sides, somewhat equally, would receive radiation from the Sun. The anti-planet-side would receive slightly more interstellar radiation, which is significantly less than from the Sun or the planet.
So, the planet side would experience the higher amount of radiation in total.
This is the general case. In certain cases, like for Europa, the radiation is lower on the side of its direction of orbit.
The radiation belts are rotating around Jupiter faster than Europa does. This results in the radiation predominantly striking the trailing hemisphere of the moon—which is always the same portion of the moon since Europa is locked in a synchronous orbit around Jupiter.
If you're willing to split into quadrants, then the leading-planet-side-quadrant is lowest, for Europa at least.