Why is the modulating shape appear in both sides of the carrier signal in AM?
First let's clarify what you are asking about.  I take your question to mean that you are looking at the carrier on a scope and notice that both the bottom and top of the amplitude envelope get modulated.  You are asking why that is done, and why not just one "side", meaning either the top or the bottom.  This has nothing to do with single-sideband modulation.

Think about what it would mean for only "one side" to be modulated.  That's just the right amount of the modulating signal added to the modulated signal.  The frequency of the modulating signal is much lower than the carrier, so is removed by anything that is narrow-band filtering around the carrier.  Now consider that is exactly what radio receivers do.

Even if a transmitter did add the modulating signal to the carrier, it wouldn't propagate out since its frequency is way too low for the antenna, and receivers would ignore it anyway.

To use typical commercial broadcast AM as example, let's say the modulating signal is 3 kHz and the carrier 1 MHz.  The antenna can't meaningfully radiate 3 kHz, and the circuit in AM radios tuned to this station go to great lengths to reject anything more than about &plusmn;10 kHz from the carrier (990 kHz to 1.01 MHz).  Even if 3 kHz got radiated, it would be completely irrelevant to the AM radios picking up the 1 MHz station.

So to answer your question more directly, modulating "both sides" of the carrier is what pure AM modulation is.  Possibly some transmitters do modulate the top of the envelope with the bottom fixed.  However, the added modulation signal on the carrier this represents is quickly eliminated by various inherent filters between the internal signal generator and the receivers.