Normally, all the probe ground/screen connections are commoned together.
They may also be connected to case / chassis / mains or USB ground, depending on the scope type.
If the scope has "differential" mode, you can use two probes (without the ground clips connected to the device being tested) and display the difference between the two, effectively giving a floating reference of anything up to the rated input voltage of the probes & scope.
Can you tie the two breadboard grounds together and bond that to the scope ground?
That would save a lot of hassle and complexities.