I had nuisance permits for my brother-in-law's and father-in-law's farms. When we got the permits, I did not have to show any proof of economic loss. But this was several years ago. Maybe now they changed the rules. At the time they handed out permits on a percentage of land being used for crops basis. It worked out to about one permit per every 50 acres. The "season" lasted from when the permits arrived (usually mid May) until the end of September. My permits were always doe only. Was not allowed to take bucks. IFAIK, all nuisance permits were anterless only but I heard some people claim they received buck permits too. Never personally saw this though.
I hunted using these permits for about 5 years or so. Took an avg of 3 deer each time. I predominantly used my shotgun but I did take my rifle once or twice. I stopped hunting nuisance permits for a couple of reasons. One: It's no fun in the summer. Simply too warm and too many bugs. Two: after killing a deer in the summer, it is a major rush to get the deer from field to freezer. Gutting in the hot weather is a challenge in itself. Fighting off flies and other bugs while gutting is a huge PITA. Then of course, it's a mad dash to get the deer processed and into the freezer asap. The third reason I don't go anymore is that it takes away from the regular season. I like building up the anticipation to opening day. If I'm hunting deer year round, that feeling is completely gone. Plus, I'd rather leave the does around to attract bucks to my land.
When I got the nuisance permits, I also got into the DMAP program. Every year, the DEC sends me roughly 10 DMAP tags just prior to the hunting season. These are basically a whole bunch of doe tags that I can use only on the designated land. I've never come close to using them all. I don't mind getting them and have used them to tag my does instead of using my regular tags. So a few seasons ago, I called them up and told them I no longer needed the summer nuisance permits. Both of my sons are now of hunting age. Between the three of us, I really don't even need the DMAP tags anymore. I wouldn't be able to fill them anyways.