If you have a somewhat newer modern monitor it may have multiple video inputs. Just plug both computers into the separate video inputs on the one monitor and then select which one you want to display from its front panel buttons..
My two 24-inch DELL monitors has many different VIDEO INPUTS all selectable from the front panel button. It also has a picture-in-picture mode which I use alot for my older M90 Dell computer. It is displayed in a small PIP view down in the bottom corner of the large screen. I can select which computer I want to go to full screen ...
I have been running LAPTOPS for years using the DUAL 24-inch DELL monitors. My laptop sits in a docking station and feeds both monitors allowing me to throw many selected views from the same LAPTOP across both monitors.
When we head out in the RV I just grab the LAPTOP and plug it into a similar setup in the RV bringing all my programs with me where ever we go...
I have a feeling however both your computers are old somewhat vintage types...
Roy Ken