Not that the National Park Service has the final answer, but the details below are what that organization has to say about this issue. I tend to agree with them. I wouldn't recommend washing hands "down by the river". Making this a common practice, especially in high traffic areas, will make a lot of unhappy, unhealthy people, flora, and fauna... Build a "wishy washy" handwashing station. Pretty simple to do, and this guy even makes it look fun:
Hand Wash Station | Hilltop Yodeler
2. Handwashing Setup.
a. A handwashing setup which involves the repeated dipping of hands into the clean water container may not be used.
b. The Service highly recommends a handwashing setup which provides for use of potable water and liquid soap in a dispenser. If this setup is used, no additional chlorine bleach or use of a gel sanitizer is required.
c. If non-potable water is used, the hand wash bucket must contain at least 100 ppm of chlorine bleach. Use chlorine test strips to determine the chlorine residual in the hand wash water.
d. Using untreated river water to wash hands, even with soap, is not a sanitary practice. All participants and guides should use treated water for hand washing. In the event hands must be washed with untreated river water, this must be followed by the use of a hand sanitizer.