I know that there's a risk of bacteria in any area where you defrost chicken, but what about the chicken itself? Will running water over it (unwrapped) reduce the taste or change the texture?
The safety of this will be related to how long the meat is in the "danger zone". If the meat temperature is within 39 degrees to 140 degrees Fahrenheit, then it is in the danger zone and should not be kept at that temperature very long.
So temperature/safety wise you will be fine thawing with water. Running water over the chicken will not change the texture. In fact, it is recommended you brine the chicken in a salt water solution before cooking.
It is common to brine a bird in a water solution so I doubt the water can hurt it.
For speed I would place it in a like a dish pan covered with water and just run a small amount of water to displace the cooled water. You would get maximum water contact for maximum heat transfer. And you would better keep the skin under 40 while thawing out the center.