Nope. But you need a lot of electrical energy to first purify the water to a certain degree. So this is surely no problem on a nuclear sub.

Electrolysis with pure salt in fact gives pure chlorine gas but also is very costly energy-wise (except you can afford some fancy, expensive catalysts).

If they'd share their technology though on youtube i can't judge.

I suppose they use classical filters and osmosis through some membrane for the first steps of purification anyways (otherwise it may take too long) and

assume they first and mostly reuse their non-salt-water.

If you are searching for a entertaining way to get into (not only water) chemistry, i suggest you get the game factorio and after mastering factorio vanilla (getting the rocket off the planet) you download two mod collections for this game: angels and bobs mods.

If you master angels petrochem you at least have a good idea how this kind of stuff tends to work.