Is this being dishonest? While my softener does use salt to regenerate the resin, doesn't it flush out the salt? I certainly don't taste saltiness with my water.
It is honest, albeit exaggerating. As stated earlier, water softeners use ion exchange resin to "soften" water. Softening is defined as removing minerals that qualify has "hard," these being mainly calcium and magnesium. The "hard" minerals are only partially soluble in water, as evident by scaling you may have noticed (especially with hot hard water, solubility of these minerals is inversely related to temperature). Sodium is very soluble in water, actually it prefers being dissolved in water over being a solid metal.
On the microscopic level imagine one or two sodium atoms attached to a long polymer chain of carbons, making an ion exchange resin. On the macroscopic level, the resin looks like very large grain sand. when magnesium or calcium come into contact with this chain, the sodium ion is replaced. Once the resin is saturated with hard minerals, it is regenerated with a sodium chloride (brine) solution. Since the concentration of sodium is much greater than that of the calcium and magnesium on the resin, these minerals leave. Eventually the resin does break down and needs to be replaced.
So yes water softeners do replace calcium/magnesium with sodium ions. If you are real concerned have the water hardness tested and as a worst case scenario assume all hardness is replaced with sodium. This will give you some idea of how much sodium you are consuming, i really doubt that it is much compared to your diet.
Typical water softeners only remove positive ions (cations). You can buy a water softener that removes both, but I haven't really looked into the resins that remove anions.
More info from wiki