Modulation takes place in the radio transmitter. Every received rf carrier from the same transmitter is modulated by the same modulator.
The local oscillator is located in the radio receiver. Every radio receiver has it`s own local oscillator but the receiver does not contain a modulator.
So, no the local oscillator is not modulated. If the local oscillator is unintentionally modulated by interference from any source then it will cause distortion in the output of the radio receiver.
The mixer amplifier in the radio receiver combines the output of the rf amplifier with the output of the local oscillator to produce a sum and difference frequency both of which retains the original modulation of the rf carrier. Therefore the local oscillator can operate at 455kHz above or below the desired rf carrier and still produce the desired difference 455kHz (if) intermediate frequency. I think it is customary in standard am radio broadcast receivers to operate the local oscillator 455kHZ below the desired rf carrier frequency.
In a typical superheterodyne radio receiver, the local oscillator, LO, is unmodulated pure sine wave. I believe that the LO is generally operating below the frequency of the incoming radio signal but can be either above or below, with the difference being the IF stage (immediate frequency).
Modulated LO, very interesting idea. This concept never crossed my mind but if done right, it might lead to some new innovations in +/- feedback gain control affecting both receiver sensitivity and selectivity. What if the detected audio signal is first attenuated and fed back to the local oscillator in the form of amplitude modulation or even frequency modulation? That might lead to some very interesting effects.
If you're talking about a superheterodyne receiver, it's not modulated, and it can be either above or below the signal frequency.
The local oscillator is tunable and is mixed with the incoming signal, which produces sum and difference signals between the local oscillator and all the signals that have made it past the input stage. This entire mess is fed to the IF or intermediate frequency stage, which is a sharply tuned circuit resonating at a fixed frequency. In AM broadcast receivers that frequency is typically 455 kHz.
The local oscillator is tuned so that the sum or the difference frequency for the signal of interest is 455 kHz, so that it's selectively passed through the IF stage and to the detector.
Many high-performance short-wave receivers do this whole thing two or three times to gain greater selectivity among other reasons, and they are called double- or triple-conversion receivers.