Asymmetrically loading barbell

On barbell, it's becomes a bit dangerous, you don't want to fail asymmetrically and drop the bar, it's a lot of weight. On a smith machine, a small weight variance is no big deal, go for it.

With dumbbells, yes you can assymmetrically load, it will greatly decrease your overall power output but increase isometric demand on abs, obliques, and spinal erectors to maintain stability. For example, some people like to do lunges carrying a dumbbell on only 1 side at a time for that kind of challenge.