That sort-of depends.
A third of a scrubbie lasts all year for me. But, I also do a certain amount of cooking with relativly large amounts of oils (Usually olive oil, vegtable oil or dried butter, ie "parified?" butter.) In a mini bottle about 1/2 full, this is good for three weeks or so. A single drop, sometimes two, on the scrubbie works well enough to cut the oils off the pot, spoon and cup for greasy dinners. Sand is used when available, or, duff along with the scrubbie. After washing, I whip the scrubbie mostly dry, it doesn't pick up any real scents, other than the water and soils in the area.
Often, sand is used, but, in the ADK's it is more often a light gravel, not really well suited to cleaning. Mud with forest duff makes a sticky mess more often than not. However, pine cones and needles have a slightly soapy effect on the pots and pans, it mixes the oils and water. I will dump a large handfull into the pot with some water and scrub it around. I reserve the soap (Dr. Bronners or the like)for final washing to keep critters out of the cookwear. In 40 years, no problems.'Corse, back in the day, I just used my bandana.
Biodegradable soap? Well, most is biodegradable. The problem is that it acts as a fertilizer in waters accilerating algae growth. Then the algae dies off and rots in the water, causing polution that way...more of a secondary effect. A lot of ponds and small lakes in NY are not even named on maps, more of runoff in granite "cups" and fairly high in acid (usually tannins.)(Acid raid has devestated a lot of these smaller sources because they were so acid to begin with.) Wash water("grey" water) is not put back into the source, but scattered over an area away from camp by at least 100'. I believe this was recomended at Baxter Park, Me., too.
In more populated areas, I dig a cat hole to bury the grey water.