It's surprising there isn't some chemical reaction to precipitate out the sodium and chloride atoms bonded to something else, or whatever.
The easier way to precipitate chloride is using silver, you mix the salted water with silver nitrate and the silver chloride goes to the bottom. But you now have a sodium nitrate solution. (It's safe to do this experiment at home, if you are careful and follow minimal laboratory safetely procedures. But don't drink the water!) https://en.wikipedia.org/wiki/Silver_chloride#Chemistry
Just assume that all the sodium salts are soluble. The only example that I remember is the sodium uranil acetate salt. But it's definitely not a good idea to drink that water later. https://en.wikipedia.org/wiki/Uranyl_acetate#Safety
[1] I think that this is the main reasons why most of the salt in the sea is sodium chloride. Related question: Why doesn't the sea have too much potassium?
To answer your question, ion exchangers do what you ask, using sulfuric acid and sodium hydroxide, and produce a ton of sodium sulfate in the resulting desalinated water.