Baudelaire, my wife and I both suffered similar swelling almost immediately after moving to our home 3 years ago, with built-in salt-based water softener. The "reaction" is not from skin being exposed to salt. It is a cardiovascular circulation and water-retention problem from DRINKING it. My wife and I are both healthy, athletic people, in good shape and 50 yrs old. It took us the longest time to figure out the problem, as we both would notice swollen ankles in the evening when we settled down for the night. I am a biologist, and it still didn't occur to me that our tap water was the problem, until one day the system's backflush malfunctioned due to a power outage, and we made a pot of coffee that was terribly salty. I started doing research on the salt-based systems, and talked to doctors about it. The culprit became obvious. Even when the system is normally working correctly, the increase in our salt intake is significant. As soon as we switched to a jug bottle of spring water for drinking, the swollen ankles went away.
While jeff72's response is technically correct about how most salt-based water softeners function, to suggest that the trace amounts of sodium chloride in the water is harmless is medically inaccurate. Generally, calcium and magnesium ions are the target of these softeners, as they make up what is called "hard" water. These metals are actually GOOD for us, but bad for pipes and appliances though. The salt softener replaces two good, healthy metals with a BAD, unhealthy molecule which is "salt". Softened water from a salt-based water softener is not recommended for drinking, watering houseplants, lawns and gardens due to its sodium content. There are many health risks associated with sodium intake. People on restricted sodium intake diets should account for increased levels of sodium in softened water. Your family physician should be consulted. Sodium intake from softened water can be avoided by leaving one kitchen tap un-softened for drinking and cooking. Not only that, but water used in recharging a water softener may over load or reduce the effectiveness of small septic or sewer systems.
Hope that helps!