Yes, you replace skimmate with fresh saltwater. If there's absolutely no color to the water in the skimmer cup, I'd be tempted to just let it overflow or dump it back into the sump - or even cut the air off and just flow water through the skimmer for a few days and try the air again later to see if the flooding has stopped.
You want your skimate to be a dark brown color. If it is light you need to adjust. You will not see much salt being removed from tank if skimmer is adjusted correctly and always use rodi for topoff.
Running your skimmate dark fouls the skimmer faster, decreasing performance between cleanings faster. Darker is not "bad", but it's not better either.
Unless you have a problem in the tank, wet skimming also doesn't remove much water - on average, I pour out a full skimmer cup every other day from my Tunze 9410, which holds about 3 cups.
Even wet skimming will not cause a drastic salinity change*, but for whatever saltwater is skimmed out you will either have to readjust tank salinity at water change time, or if you can keep some made, add back fresh saltwater at the time you empty the skimmer cup so you can measure and add it back 1:1. (Just refill the skimmer cup with saltwater equal to the skimmate you cleaned out.) The later will give the tank the most stability, but it's obviously going to make emptying the skimmer a little more work. I do water changes every day, so it's no problem for me to adjust it at water change time and still have better stability. Whatever works.
I think in the old days when ATO's were much less common and everything was much more expensive it made a lot more sense to skim dark and dry and just put the extra elbow grease into keeping the skimmer clean. (Look at the height on some of the old DIY skimmers!!) I think in most situations, folks are best off skimming fairly wet to keep the skimmer cleaner longer. (Most people aren't going to scrub the skimmer often enough anyway, so why exacerbate this? It kills skimmer performance.) Dealing with a little extra salinity creep isn't that hard in most cases.
-Matt
* If your setup allows your skimmer to remove 5 gallons of skimmate and your ATO to replace 5 gallons of missing water, you are by design at risk of that happening at some point.

Having a smaller skimmer cup that just floods back into the sump has some advantage.
