The short answer is yes! When you use a bar of soap (or Tooth Suds) the oil attracting end of the soap molecule picks up the grease and oils on your skin. When you rinse, the water-attracting end of the soap molecule follows the water, rinsing the soap and any impurities down the drain when you wash it off (or spit it out).
After washing or brushing your teeth, any bacterium that was present on either the soap or your hands/teeth is washed down the drain.
Here's more information about why soap isn't dirty and perfectly hygienic and sanitary.