We just got a nebulizer for the first time at our clinic. Our respiratory therapist told us to wash with warm/hot tap water and dish soap, rinse them with tap water, and then use a baby bottle sterilizer with distilled water to sterilize the pieces. The sterilization should kill anything that the tap water left on them after it was washed.
She said our other option is to wash them with tap water and dish soap, rinse, and then bring a pot of water to boiling, drop in the pieces, and let the water be at a rolling boil for 5 minutes.