Why do carrier waves sound silent on radios?
Your question describes a situation which, if accurate, goes a bit beyond a basic receiver.

First, the background:


  I was recently reading about and building low-power AM radio transmitters, and read/noticed that when the carrier wave is not yet modulated, the radio receiver goes silent.


As you're probably away, an AM envelope detector demodulates only modulation, and does not produce sound from the mere presence of a carrier.  This is not, however, the only type of detector in use - a product detector especially in the form of a tunable beat frequency oscillator can be tuned to produce an audible tone in the presence of a carrier alone, and is how both morse-code "CW" signals are demodulated (via a frequency difference) and how single sideband signals are demodulated (by matching the original carrier, or nearly enough for voice).  It can also be used to demodulate ordinary AM, but unless the received is carefully tuned, the carrier will be audible as a fixed tone of nonzero frequency.


  For example, let's say my transmitter transmits at 1000 KHz. Before the transmitter is turned on, the receiver picks up the local radio station that also transmits at 1000 KHz. However, once my transmitter is powered on (but not modulated), the radio receiver becomes silent.


On the contrary, this is not something that actually occurs with a basic AM receiver.  Such a "capture effect" is something that occurs with an FM one, but the lack of this behavior in a basic AM system is famously the reason that aviation communications are done with AM - the ability to "talk over" other transmitters and and have both transmission be intelligible as if this happened acoustically in a room is considered a key safety factor.

Instead, it is likely that what you have is an automatic gain control, a system where the strength of the signal causes the receiver to turn down its own gain.  In that situation, then a strong signal would indeed quiet weaker ones, as if a robotic hand had reached out of the chassis and adjusted the volume knob.  And AGC is typically driven by the strength of the carrier, not the volume of the audible modulation.

But this would be an intentional feature.  Non-linear distortion such as compression when stages are overloaded does not really have this effect, because  independent signals will not line up only at the waveform peaks, but rather the weak one will also be present in the troughs of the strong.  There will be distortion generating additional intermodulation frequencies, but the original signals will still be there as well.