I saw a bathroom scale that also tells you your body fat percentage. Does this thing work?

Aside from putting the inventor's kids through college, no.

The "scale" uses a method called bioelectric impedance (BI). BI calculates body fat by measuring how electric current passes through the body. Proper BI uses electrodes attached to a leg and an arm that enable the electric current to pass through the entire body, including the trunk. This is important as most adults have a different distribution of muscle and fat between the upper and lower body. In a BI "scale," the electrodes are only under the two feet of the upright individual. The current may not go effectively through the upper body, which can skew the accuracy of the body fat reading.

One of my clients used this "scale" and thought she had 45% body fat. I measured her body fat with skinfold calipers and determined that she had 27% body fat. The "scale" gives skewed readings particularly for women, as women tend to carry extra fat in the lower body.