OBS's audio is terrible. The quality is there, but the tools are some of the hardest to use that I've ever seen. In this particular case, the near-complete lack of metering and lack of transparency as to *where* each thing actually connects, means that you're pretty much in the dark as to what the "correct" settings really are. Compare that to a DAW plugin:
That's also a side-chainable compressor (side-chain source and settings on the bottom left, compressor settings to the right of that), with metering and graphs all over it so you can see what's actually going on!
For this one in particular, the left graph is a nice history of what it saw and what it did over the last 5 seconds, and the right graph is what it's doing right now. You can see the classic compression curve, and the dot "dances" along that curve to show you exactly where you are on it. So if you want to "ride the knee", for example, you simply adjust the Threshold with your eyes to put the dot on that part of the graph.
Note: That does NOT mean that you should mix with your eyes! Always mix with your ears, and *then* look at how things ended up. If you want to copy something, then you copy it by eye and then check it by ear.
I suspect in your case, that your side-chain source is below the compressor's threshold. But with hardly any metering in OBS, it's hard to tell.
Or it could be that OBS's Monitor tap is pre-filters. Again, lack of transparency.
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Also, if you're using the side-chain, then you're right on the edge of usefulness in OBS anyway. It won't do much more. So I'd also look at moving all of your audio work out of OBS and into a DAW, and then pipe the finished soundtrack from the DAW into OBS as its only audio source at all, to pass through completely unchanged. Your headphones, then, would come from the DAW, not OBS.