To the best of my knowledge, those body shaper garments just make you look better while you're wearing them, they don't actually change your body composition.
Actual proper "waist training" with a heavy duty corset CAN reduce your waist, but not by reducing fat but rather by causing your organs to shift. And you have to keep wearing the corset regularly to keep the results, which is a chore and a commitment. I'd personally rather save up for liposuction than mess with all that trouble, expense,discomfort, and ongoing maintenance.
The other problem is that wearing a corset or even those body shaping garments over long periods of time actually causes your core muscles to atrophy, since the garment holds you up and reduces the body's need to hold itself up.
hope that helps.