Eventually, the plumbing trap will fill with the sediment of all the clay and ceramic materials being swept down into it. This will happen fastest in the first container of a two-stage trap, but both sides should be cleaned out periodically.
When the sediment is within five to eight inches of the pipe connecting the two containers together, the trap must be cleaned. To do this, I recommend using a very large (institutional sized) ladle. I strongly suggest wearing rubber gloves, as well. Microbes have been finding a fine home in the sludge.
Scoop the sediment out and into a bucket. Get as much sediment out of the trap as possible, then allow the sediment to settle again before using the sink attached to the trap. This will help keep particulates from going down into your main plumbing.
The sludge you have pulled out will almost certainly stink from the microbes in it. If the sink is mainly used in washing away clay, you can dry out the sludge and use it. Since it may be unsavory to handle as well as being a mixed bag as to characteristics, I'd suggest using it in projects where you don't have to handle the clay much, such as plain (at least until after bisquing) tiles.
If the sludge has a fair percentage of glaze components in it, dry it out then fire it inside a throw-away bowl or vessel (in case it fuses). Once fired, it can be sent to a landfill with much less concern about potential leaching into the ground water.