Measuring RF and heterodyned frequencies
The reason you cannot see the other signals is likely because the voltage waveform of the local oscillator is several (6-8) orders of magnitude greater than the receive AM waveforms. Whether your oscilloscope is sensitive enough to make mili-microvolt measurements will determine whether you can "see" these receive waveforms. There should be another amplifier stage after the mixer, at which you may be able to see the sum/carrier signals. Without seeing the rest of the schematic I can't tell you if there is a second IF stage of if a diode/BJT detector is used to restore the envelope of the down-converted signal. If there is another mixer stage you will be able to see the sum/difference signals at the output of that stage more easily.

Edit: After looking through the datasheet for your oscilloscope, you could also take the FFT of the probe signal at Q7. From looking through the datasheet it seems like the voltage measurements are only accurate down to around 2 mV, so I don't think you will be able to measure anything in the uV range.