Is the private organ of a man or woman considered najis (impure), such that it would defile whatever it touches, be it clothing or anything else?

The private parts of a man or a woman are pure, based on the Prophet's (peace be upon him) saying: "Indeed, a believer does not become impure." So, if the private part touches something pure, it does not make it impure, unless there was impurity on it that transferred to the touched spot.