Try switching the electrics off. If you can see a swing on one, and not on the other then I'd suspect the moving one. Track a radial or ideally a LOC beam. With some idea of the wind (HW or TW ideally) check which compass reading compares favourably.
If in good VMC measure the bearing between a pinpoint you're over and a visible point as far away as possible. Point the a/c to the vis. point and read both compasses. Which one is more accurate?
Also make sure you're checking & comparing using the deviation card and not the raw compass heading.