Ours tend to bring lovely little presents, like barn rats. Bimmer eats anything else he hunts, and so far Shiva and Kharma haven't caught anything else - that I know of! (lol)
When our first Fila, and the love of Bimmer's life, Buffy, was poisoned and died, he started stealing stuffed toys that were the same color as Buffy. The first one was even a stuffed dog that kind of looked like her! When Shiva went through her first cycle recently and we had to keep them separated, he stole another toy - the same color as Shiva (and Buffy). I have no idea where he gets them.
Shiva basically steals anything that takes her fancy and brings it in. The last thing she brought in was someone's air conditioner cover! We're out here on 60+ acres, but there's a trailer park next door. They never used to go over there, but now they have a friend, Stanley, the black Lab puppy, who lives over there and they all visit back and forth. I really don't like them leaving our property, but Stanley comes over here, then they go over to his house - just like kids. I guess you could say they're thick as thieves!