There were never 20,000 ancestral puebloans living in just the canyons of Mesa Verde. However, to quickly answer the question, the inhabitants who did live there used latrines to dispose of human waste, often situated in leftover rooms and structures. This was manageable because any particular site in the Mesa Verde region (this term is misleading) had no more than a few hundred people living there at a time [1]. For basic toiletries, they probably used juniper bark or similar. There has also been speculation that at least some puebloans made soap from Yucca roots, though this is unsubstantiated to the best of my knowledge.

Let's return to the phrase Mesa Verde Region, which is probably where the confusion is originating from. Despite the name, the term encompasses far more than just the park we call Mesa Verde today. Here's a map of the area from the crow canyon archaeological center. This entire area, including large parts of New Mexico and Utah, had a population of approximately 20k (see the VEP estimates in [2]). This region had many waterways and an average population density about the same as it is today (i.e. not much).

[1] Mahoney, N. M., Adler, M. A., & Kendrick, J. W. (2000). The changing scale and configuration of Mesa Verde communities. Kiva, 66(1), 67-90. https://doi.org/10.1080/00231940.2000.11758422

[2] Varien, M. D., Ortman, S. G., Kohler, T. A., Glowacki, D. M., & Johnson, C. D. (2007). Historical ecology in the Mesa Verde region: results from the Village Ecodynamics Project. American Antiquity, 273-299.