If you want to do this with high quality video encoding then I would suggest buying a capture card that supports hardware encoding if you want to offload to the old PC. You can duplicate the monitor (or use obs preview on the capture device) with very, very little overhead on your gaming pc, and on the core 2 machine, the encoding on OBS and out to your streaming platform will be handled by the capture card.
I do this between my main rig and an old AMD FX machine and stream 1080P 30FPS (all my specific capture card supports, got it for $50 on ebay..), very low load on the CPU, Q9550 should be just fine.
Another alternative would be to buy a used graphics card that supports x264 encoding and stick that in your main rig, set OBS to use that device for hardware encoding. I've never tried this route as I wanted all work to be completely offloaded to another machine.
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Last edited by Serious_Don; 03-09-2019 at 09:53 PM.