I made a current sensor using a 741 op amp, as a differential amplifier.
I tried putting the sense resistor on the positive side of the load (as in your schematic). However I found it performed better when I put the sense resistor on the ground side of the load. Reading minute voltages could be done more consistently when the input was referenced to ground.
My sense resistor consisted of several inches of 14 ga. wire. It had to carry several amps. I made it just high enough resistance so that it would generate a measurable voltage for the op amp to detect. After some experimentation I found a suitable voltage was several mV when maximum current was going through it.
I amplified it several hundred times so that the output gave me a range between 0.1 V and 9V.