We need to distinguish between different ways of looking at a signal. On an oscilloscope, a sine wave will look like this:
But if we look at that on a spectrum analyzer, it'll look something like this:
...a little noise (mostly from sampling and such) but mostly one sharp peak at the sine wave's frequency. If we synthesize data, run it through an FFT, and graph the input and output, the result is more "perfect"--a single spike at one precise frequency, with none of the noise:
Here the axis/waveform is green, and the FFT is that single, vertical black line.
When we amplitude modulate the sine wave, we mix the audio signal with the radio signal. In the process of mixing the audio signal with the carrier, we not only modulate the carrier, but also produce two sidebands: one at the carrier frequency + the audio frequency, and one at the carrier - the audio frequency.
So, we look at this on the spectrum analyzer, we get something like this:
The tallest spike at the center is the carrier frequency, and the two others are the side bands (the one to the left, at the lower frequency, is the lower side band, and the one to the right at the higher frequency is the upper side band).
If we synthesize a modulated wave form by simply generating 3 perfect sine waves at slightly different frequencies, and then run them through an FFT, we get a rather cleaner display:
Either way, what we have is three sine waves all being transmitted together--but each is still just a normal sine wave like that shown in the first picture above. The only thing unusual involved is that they're at three slightly different frequencies (and as we see in the amplitude plot, the result no longer looks like a sine wave).
Putting them through a diode won't change that basic fact. We'll just have three sine waves at different frequencies, but with all the negative (or all the positive, depending on the diode's orientation) removed. So each of the sine waves will look like this:
...but we'll still have the upper halves of the sine waves at all three frequencies, so on the spectrum analyzer it'll still look roughly the same as it did without the diode. It does change a little bit (because rectifying it obviously changes the shape of the waveform) but not drastically.
Oh, we could also use a full wave rectifier, giving an output roughly like this:
If we run a modulated wave form through a diode, then an FFT, we do see some artifacts from the rectification. For example, with three sine waves at relative frequencies of 0.7, 1.0, and 1.3, we get a result like this:
So at this point, we get a fundamental, two sidebands, and some artifacts from the rectification--the sudden change of direction where it's dropping to zero, then starts to rise again acts like some other frequency of signal for the duration of that transition (and the apparent frequency of an artifact will depend on the rate of change on each side).
Getting to the exact question you asked: it depends on what you mean by AM radio. If you mean the commercial AM broadcast band (540 kHz up to 1700 kHz), then the answer is that all signals there were and are broadcast with the carrier and both sidebands.
On the other hand, the carrier uses a lot of power, that carries essentially no information. If you use a narrow band-pass filter right after the modulator, you can filter out the carrier and one of the side bands. By transmitting only one side band, a given amount of total output power can result in considerably greater range. You can also optimize bandwidth usage, by allowing receivers to tune in only a lower side band or an upper side band at one time, so each can act as essentially a separate channel.
Although (as stated above) this hasn't been done in commercial AM radio, it has (for one example) been done in CB radios.