We would typically process stereo submixes and master mixes using regular stereo processors. And in the case of a compressor, although the left and right channels could be processed completely independently, our compressor usually has a stereo linking mode that uses a combined left and right channel sidechain so that the stereo image remains stable.
However, you’ll often hear mixing and mastering engineers talk about processing stereo signals using the mid/side technique. This splits the signal into its sum and difference components allowing you to process them independently and then returns them to stereo at the output. Much like recording in MS, this process only requires two audio channels, which means that it’s reasonably straightforward to design a stereo compressor that can also process in mid/side (the classic Fairchild 670 for one example).