There is slight crosstalk introduced. However, this is usually little enough to be negligible in actual consumer use.
If the ground wire were an 'ideal ground', that is absolutely zero resistance and inductance, there would be no crosstalk. That's because although the ground wire carries the current for both channels, because it's a perfect ground it always has zero volts at each end, so each of the left and right channel can reference an ideal 0 volt ground.
In practice, the ground wire has some resistance, let's say it has 0.1 ohms. If that cable is feeding a pair of 32 ohm headphones, that ratio is 320:1, or roughly 50dB, approximately the level of the crosstalk that would be generated. That is a better channel separation than vinyl records or multiplexed FM ever achieve, and once you put two loudspeakers in the same room, is anybody really worried about better than even 10dB?
If that cable were feeding another amplifier, rather than headphones, the input impedance would be much higher than 32ohms, and the ratio better, that is, practically perfect.
In studio work, all signals are carried individually.