For example, if the levels on the left channel are above the threshold and the right channel is below it, will it still attenuate the right channel?
Yes, it will. This is a regular stereo compressor which will treat both sides equally once a signal surpasses the threshold. This way, the stereo image is kept as is and sort of glued.
There are dedicated Compressors like the famous Fairchild 670 which can be operated in Dual Mono mode. In this case, each channel is treated separatly which results in a broader stereo image or adds a more “lively” feel to it given it’s not overdone. At the same time it might weaken the center so it’s a double-edged sword.
I’m not sure if it’s a double-edged sword - I have double-tracked guitars, and I don’t want to copy and paste compressor settings or link inserts between the two channels. But I guess I get to.
Anyway, thanks for the answer!