I think the answer depends upon how wide and rigid the base of the drawstring bag is as well as your body shape and proportions. If a drawstring bag hits you above the waist or if you have narrow hips, it will stick out less than if it hits at the hips and those hips have a little padding of their own.Do you find that drawstrings stick out too far from your body or do they carry well?
Also, some ladies wear the drawstring handbag slung to the back of their body and others wear it on the side.