Like other users pointed out, pickles still have calories, just negligible enough that the manufacturer can label them as 0 cal (I think <5 cal/serving). However, if you ate 1000 of them, those so-called negligible calories would add up.
Same idea with sugarless gum.
Also, I ate a (delicious) Jimmy John's pickle today, and their nutrition website list the calorie content as 15. It was a big pickle.
the simple act of chewing and digesting the pickles negates any calories that were in them to begin with
This is pretty much the same idea as so-called "negative food" (calories taken in are equal to or less than calories burned by chewing/digesting). I've read in more than a few places that the math there doesn't add up. For one thing, average calories burned chewing is 11 calories per hour. If it takes you more than an hour to eat a pickle, then you might have other things to worry about. I think there might be a sticky somewhere in the forum that talks about this, as well as calories burned during digestion in greater detail.
A few pickles probably won't matter - they might as well be 0 cal. But 1000 pickles will add a lot of calories. Also, you'll have brine breath, an urpy tummy full of pickles, and be holding a lot of water weight from so much salt. I bet you'd fart a lot, too.
