We've done this using several different methods:
1) If both systems are on the same fast network:
- install NewTek's NDI tools on the "primary" computer,
- use their "NDI Scan Converter" to capture the display screen
- Install the OBS NDI support extension
- Set up a scene with the NDI Source in OBS
2) if no network is available
- Use a Decimator MD-HX as a HDMI splitter and SDI relay between the primary computer and the primary display
- Run a SDI coax cable to the OBS computer
- Use a mini recorder (or quad deck link or ...) to bring in the second stream into a scene...
3) We've tried #2 above, but using BMI's HDMI-SDI box, both with and without a HDMI splitter instead of the Decimator, and ran into hard to reproduce/fix issues with HDMI computer monitor timing sometimes not being accepted by the BMI gear. The symptom was no video being "seen" by the mini recorder... Sometimes the order of powering-on the dongles mattered, other times cable disconnects during a service caused things to stop responding, even after reconnection. When it worked, it just worked - but swapping a laptop or changing screen resolutions or a particularly book-thumping sermon, and poof! :-)