If you were being served on dirty dishes in medieval Europe, you'd best think twice about the quality of the establishment. Medieval people might not have understood the concept of "antibacterial" or "sanitization," but in general they placed a high premium on cleanliness--especially in the context of hospitality. A standard part of the place setting in polite company was a dish to use to wash one's hands off. Visitors to an upper class house or castle would be greeted with a tub to wash off their hands and faces. So yes, they washed their dishes.

Water would likely come from a well or one of the town's fountains, if it had them. In some cases people hauled water from a river or a creek, although they understood that some rivers' water was unsafe to drink unless it was boiled.

The major ingredient in medieval soap was wood or vine ash, which would be mixed with ingredients like oil, eggs, alum, and perhaps something nice-smelling to form either a "hard" or a "soft" soap. Generally, "hard soap"--I'm thinking more like bars or cakes here--was a Mediterranean-region product and soft soap was northern, although rich northerners sought out hard soap.

Dish-washer was a dedicated job in at least some places. According to Jacques Le Goff, Thomas Aquinas considered it one of the worst jobs to have--philosophically speaking--because of the close association with the unclean.