washing dishes-is grease and food safe down sink drain?

actually, the more I think about it, its sooo cheap to buy paper plates,cups, utensils its probably easier to just throw plates out... and I think im gonna buy a small plastic tub and just wash any utensils or cooking grids,right under the outside shower with soap water and just find a place to dump the soapy water. im not sure if this is legal or not at campgrounds? im sure alittle soapy water dumped in the dirt isn't harming anything?