It doesn't need a common ground, the circuit left of Rs just needs some type of reference so it doesn't float.
If it has some kind of reference so that it does not float, then in practice is has a common ground whether two "grounds" are explicitly connected or not.
With that in mind, what happens in this case if there is no connection? See below.
In below diagram; Do I need to have a common ground between V1 and V2 or not?
If V1 is truly floating, then ground on the right charges to the common mode voltage at the inputs and that becomes the reference between the two circuits. This can work however it can also present a safety hazard if the common mode voltage is high compared to Earth ground so it is discouraged. See below about how multimeters work.
The above is why high voltage differential oscilloscope probes should *never* be used with an isolated oscilloscope input. If they are, then the BNC shield side of the isolated oscilloscope input will charge to the common mode voltage at the input to the differential probe, which can easily be kilovolts or 10s of kilovolts if that is the sort of measurement being made, which is both hazardous and likely to damage the isolated input. For safety reasons, high voltage differential probes have three inputs with the third being the common reference at their BNC shield output.
How do the multimeters measure the voltage? with an in-amp?
Multimeters, with the exception of differential multimeters, make a singled ended measurement with common referenced to one terminal, usually the negative terminal. The measurement circuits are galvanically isolated from everything else which allows the multimeter's common connection to charge to the input voltage. This asymmetry between the two input terminals is reflected in the specifications which list a "common mode" input capacitance which is the capacitance from the negative terminal to Earth ground, often between 100 and 200 picofarads.
This asymmetry in the input impedance of the two inputs can cause problems with some types of high impedance measurements so a multimeter with true differential inputs would be required. Or for instance connecting the ground lead to a sensitive circuit node could cause unintended operation with its effective 100 to 200 picofarads of capacitance to Earth ground.
For a handheld multimeter, the entire unit is contained and insulated allowing the common internal connection to charge to the voltage applied ot the negative or common terminal. Bench multimeters use an internal isolation barrier for power and usually data to the measurement circuits so the negative "common" terminal is floating.
You can verify the above on a handheld multimeter. Hold it in one hand and measure line voltage with *one lead* to the common terminal and then the volts terminal. The volts input will show a few volts, measured to the multimeter chassis which is capacitively coupled to your hand and then Earth, while the common input will show almost nothing because it is connected to internal common and the chassis so there is nothing to measure.
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