I don't have much to say about this. I've not tried it, and there doesn't seem to be a lot out on the 'net about it -- just a mention here and there. The impression I get is the salt brine is used to make the soap paste, but I suspect water is used to dilute the paste.
The primary fat should be a high oleic fat -- olive, HO sunflower or safflower, etc. This type of fat is going to respond to the salt to add thickness. If you don't have a high % of oleic acid in the recipe, the salt is not going to thicken the soap much if at all. One seller says her recipe has 60% olive oil with the balance being coconut and castor.
For the secondary fat, I'd use a high % of coconut oil -- at least 30% or so. That is going to help create bubbles in salty water. I might include a dab of castor, but I think I'd avoid any high stearic-palmitic fat like cocoa butter, shea, palm, etc.
Salt will have two contrary effects on a liquid soap. Salt will thicken the soap up to a point, as long as the recipe has sufficient oleic acid. When the % of salt becomes sufficiently high, the reverse to happen -- the soap will become thinner. That's part of the reason why I don't think brine is used for dilution; just to make the paste.