In a hunting scenario, this is a pretty easy answer: It’s important to pick up and pack out any trash you generated while in the field, and that includes your shotgun shells. This is usually not a big deal in a duck blind or on a dove hunt, where your empties will end up in a relatively small, contained area and can be easily gathered at the end of the hunt. If you’re moving, like on an upland or small-game hunt, and you’re shooting a pump gun or a semi-automatic that spits shells several feet away, it might be more of a challenge, but do the best you can to find your empties and carry them out of the field. A shell-catcher can help you keep track of your hulls if you shoot a semi.
The situation changes a little when it comes to competitive or recreational shotgun sports, and the answer is: Ask the club. At most clubs, it’s a rule or at least considered standard practice to pick up after yourself, and that means gathering all of your spent hulls and disposing of them. On the sporting clays course, you do this at the end of each station. On the skeet or trap field, it’s often done all at once when the entire round has been completed. Ask club management or members what’s expected.
On the other hand, some clubs have a “If it hits the ground, it’s ours” rule regarding spent shells, and club staff gathers the empties regularly to dispose of, recycle or reload. You need to know if this applies where you shoot. (Note: This also applies to spent brass and/or live ammo on some handgun and rifle ranges, where management fears you might accidentally pick up a live round someone else dropped and cause a safety hazard by using it in a firearm for which it was not intended.)
Be aware that whatever your club’s rules are, they might change during tournaments, where picking up hulls can slow a squad down and cause the entire event to run behind. If you’re shooting a tournament where they don’t want you to spend time picking up your spent shells, shoot management will generally let you know this before shooting begins.
Bottom line: Know the rules at your club, but unless otherwise stated, assume a club and your fellow shooters would appreciate you picking up after yourself and disposing of your empty shotgun shells when you are finished shooting. If you’d like to gather shells left on the ground or in the bin by other shooters (reloaders are notorious for this), be sure the club is OK with it before you go holding things up by rooting through the trash looking for your preferred brand.