(Followup to this post from November)
This is a diagram that I found on the Intertubes of a multi-stage distillation unit. The principle is the same for Navy ones.
Seawater comes in (though I don't recall chemicals being added). Note that the incoming seawater is fed through coils of piping inside each of the chambers. The water that had flashed to steam condenses on the outside of the coils and then drips into the collection pans. In so doing, the seawater coming into the distillation unit pick up a little bit of heat. Each successive stage of the unit is at a lower internal pressure, which means steam condenses at a lower temperature, which is also why the seawater/cooling water lines run opposite to the flow of the brine in the condenser.
After picking up some heat in the condensation coils, the seawater coming in is heated to near-boiling. The steam ejector, shown on the upper right corner, is used to draw a partial vacuum in each chamber. As I described in the earlier post, the hot seawater is pumped into the first stage, where some of it flashes to steam. The steam condenses, the condensate is collected, and the now-slightly-briny water goes to the second stage and the third stage. Each successive stage has more of a vacuum, the brine boils at lower and lower temperatures and more fresh water is made.
The output of the ejector is contaminated somewhat with salt, so the steam waste is not recovered. The brine exiting the distillation unit is pumped overboard.