Your water does not have salt in it when using a water softener powered by salt. It is about the ions in the water shifted using salt as a catalyst.
"Water softeners operate on a simple principle: Calcium and magnesium ions in the water switch places with more desirable ions, usually sodium. The exchange eliminates both of the problems of hard water because sodium doesn't precipitate out in pipes or react badly with soap. The amount of sodium this process adds to your water is quite small -- less than 12.5 milligrams per 8-ounce (237-milliliter) glass, well below the standard set by the Food and Drug Administration for "very low sodium"
All of that taken in to account, does the small amount matter to clay, or say even watercolor, or other types of work where water is a component? I don't know, so maybe experimentation is needed here. As I have said in other posts, I found that water effected the color intensity and life of paint in watercolor. I don't know what I was doing wrong or right, but that changing from tap to bottled water worked.