How to analyze analog circuits with transistors [is this circuit immediately analyzable to anyone]?
Yes, the left two transtors each form a oscillator with their particular coil.  The two signals are then added.  This will cause the amplitude of the signal to go up and down at the beat frequency, which is the frequency difference between the two oscillators.

The oscillator signal with amplitude variations of the beat frequency is "detected" by the circuit around Q3.  This makes the signal that is roughly the amplitude envelope of the beating oscillator signal.  This is then gained up, rather crudely, by the remaining transistors to eventually drive the speaker.

The system functions by the metal you are trying to detect changing the inductance of one of the coils but not the other.  The absolute frequency change is small, but it is made much more obvious by causing a beat frequency between the disturbed coil and the reference coil.