Your plumbing hates clay...or perhaps it loves it. So much so, that it will keep clay and other ceramic materials in the pipes forever. Eventually, clay will cause a clog that nothing short of removing pipes and physically cleaning or replacing them will get rid of. Remember, clay is a sedimentary mineral, and it will lay down sedimentary layers in your plumbing.
You, as a potter, need water available in your studio as you work. You also must consider how to deal with waste water. Dumping it down a regular sink, floor drain, or toilet is the worst option and will almost certain cost you money and frustration later. Also, dumping it outside into the grass is not a good option. Ceramic materials can be hazardous and should not be introduced into your ground water. In addition, the clay will clump; plants have a hard time breaking through it, which can result in brown spots on your lawn.
The simplest way to deal with the problem is to use a series of buckets. Think of it as a three-step rinsing process. The first bucket is for the rinsing off of dirty hands and tools. The majority of clay (and other ceramic) particles will settle to the bottom, but some clay will remain on really dirty hands and tools. Use the second rinse bucket to get most of the remaining clay off, and a third bucket to ensure that everything is as clean as possible.
As the water gets dirtier, move the first bucket out of the rinse line. It can be used to recycle clay or set outside to evaporate the excess water out. The solids that remain can be added to your clay scraps for recycling.
The second bucket moves to the number one position, the third becomes number two, and a fresh bucket of water is added to the number three slot. If you are comfortable doing some real plumbing, however, you can make your own plumbing trap....