Left unchecked hogs, like most any other animal, can't start to cause damage. I don't have a problem with a land owner killing off hogs to manage his property.
I have family in south Mississippi and they have started having problems with hogs getting in their fields, they do more "property damage" than "crop" damage. They shoot everyone they see, process what they need for family, give away what they can, and leave the rest to lay.
Tried letting other people hunt their property to help kill off more, but like most privileges they were soon abused so they stopped letting other hunt.
Hogs have been here in the North Georgia Mountains for years, there are pockets of areas where they are starting to damage land. You can shoot them year round on private land, and on Government land anytime there is a season open. I'd like to see that changes to year round on Government land also.