ddbis, you are on a sodium-restricted diet. Table salt is sodium chloride, so it is one of the things that can affect your sodium intake. Nutrition labels list sodium levels so you can determine the contribution of other things that add to that intake. The lables don't list the chloride part of table salt because it is "just along for the ride".
When table salt or any other form of sodium chloride regenerates a softener, sodium knocks the hardness off the resin and replaces it. The chloride is rinsed to drain along with the hardness and excess sodium. When hard water flows through the resin, hardness is removed and replaced with sodium, not salt.
To estimate the increase in the sodium level (ppm) after softening, multiply the hardness level (grains per gallon) by 8. For example, if the hardness level is 20 gpg, the sodium level will increase by 160 ppm. If you consume the national average of about 1/2 gallon per day, that number doubles to 320 mg additional daily sodium intake. Keep in mind none of this includes the sodium already in the hard water.
As Clay noted, potassium chloride can be used in place of sodium chloride to regenerate a softener. However, the potassium level will increase by an amount equal to the hardness level times 13.9. Elevated potassium intake can be a problem for those with impaired kidney function.