I have a puppet with a Left Quarter view as its default. Both sets of eyelids react to the camera movement, but the right set reacts at about three times the strength. I've examined tags, I also tried removing the "left quarter" and "frontal" tags, but no luck.
The eyelids on this puppet had three variations for a swap-set and I was able to get a working puppet by getting rid of the options and it worked fine. So then I tried making full eye folders for each position and making the eyes a swap set and I got the same weird thing where only the right eyelids overreact. I'm finding workarounds here, so I'm not screwed, but I would LOVE to know why this happens and if I'm the only one it's happened to.
Can you share a screen shot of the rigging and any handles on the layer? The strength can be affected by things like how far apart handles are. Eye Pupils (not eyelids) are affected by the "pupil range" layers for the left and right eyes - the bigger the area the pupil is allowed to move in, the more sensitive the movement. Without seeing the artwork or rigging hierarchy, I would guess the pupil range is different for the two eyes.