I just bought a kitchen sink from STS (it is "just" a bag) which says to use it at least 100 meters from water. Since one can use the sink to do anything the distance seems valid, but I wonder -- if one has only a shirt with dirt/sweat to wash, and the soap is biodegradable (Dr.Bronner's) is it still 100 meters or the safe distance is smaller in such case?
It is the soap that is the problem, not the the dirt washed out.
This site Even Biodegradable Soap Can Pollute Water Sources says
We really need to clear up some misconceptions about biodegradable soap. There are many people out there with good intentions who are using biodegradable camp soap or shampoo when camping or backpacking thinking that it is good for the environment. Biodegradable soaps are preferred over other soaps but should still be used at least 200 feet from any water sources.
and on to this (my bolding)
Biodegradable soap is NOT biodegradable when it ends up in a river or lake because it requires soil for it to breakdown properly.
Biodegradable soap is NOT NATURAL.... it is a chemical.
If you wouldn't want to drink it then keep it out of water sources.
The site has other advice about alternatives to so-called safe soap.
The website of 99Boulders talks about biodegradable soaps:
Do not dispose of any soapy water in backcountry water sources — even water with biodegradable soap.
Instead, to dispose of soapy water, most biodegradable soap brands and Leave No Trace principles recommend you pour it into a cathole that is 6 to 8 inches deep and 200 feet away from water sources.
Doing so has a couple benefits:
It lessens the attraction of wildlife as compared to the “broadcast method” which also broadcasts the smell.
It allows bacteria in the soil to biodegrade the soap.
Exploring what it means for something to be biodegradable, the Balance Small Business website about environmentally sustainable businesses, says:
If something is biodegradable, then, given the right conditions and presence of microorganisms, fungi, or bacteria, it will eventually break down to its basic components and blend back in with the earth.
So being biodegradable, does not mean the product is harmless, or can be disposed of anywhere. It means that, in time, in the right conditions, the substance will decay in a way that is not harmful.
First, I agree with @Weather Vane that "it is the soap that is the problem not the dirt washed out." Carrying that further, unless you are going directly to a formal or business function from the trailhead (unlikely!!) you can get a shirt clean enough so that it feels crisp and smells clean and looks pretty good without using soap at all.
This works best if you are near a small waterfall. Just dip the shirt in the bubbly, highly oxygenated part of the stream and rub the dirtiest parts. After a short time, the shirt will be clean enough to feel clean. Sorry, I have never measured the time, because I get mesmerized by the flowing stream. 10 min should be more than enough.