How does this telephone circuit eliminate the sidetone?
Just an intuition, I didn't do a complete analysis:

The first transistor is a common collector and amplifies the microphone signal

The second transistor amplifies the line signal but since it has the collector driven by the first emitter it has the microphone signal subtracted (also, it drives the line with the microphone signal)


I also have a feeling that the 4k7 resistor between the microphone and the first emitter gives some feedback but I'm not sure of that
I'm interested if someone has a better analysis