Sometimes images are worth a thousand words, so let’s look at some pictures of a middle-side compressor behavior.
A middle-side compressor like ATKStereoCompressor can work in a middle-side workflow. This means that the stereo signal is split in a center/middle channel and a side (L-R) channel. Then each channel is processed through the compressor independently and then recreated after:

So let’s take a stereo signal (I won’t extract the middle/side channels, it is easy to get them from AudioTK):
From this, it is easy to generate the RMS power for each channel:
Now, after an attack-release filter, we can apply the gain stage with the desired ratio, knee…
The gain processed for the two channels are similar, following the same trend, but with vastly different features.
Finally, we can apply the gain on the middle and side channels before regenerating the stereo channels and get the following result:
Of course, changing attack/release values will also change the shape of the signals, as well as the ratio: with a higher ratio on the side, the signal will sound more like a mono signal, whereas a higher ratio on the middle, the stereo image will be broadened.
The code to generate these images for any stereo signal is available on github.