For ANY new toys, mine both always pick one that is "better" (even if they are identical) and they both want that one. In addition, my male is "top dog" and he will steal all the new toys and even lay on top of them to keep the female from getting them.
Since I have just two dogs, it's easy enough for me to keep things under control. With more dogs it becomes more problematic.
With lower valued toys (say, a cardboard box) I let them sort it out and they invariably end up swapping, where each has the thing that was given to the other dog. WIth highly valued toys, like squeaky toys, I try to keep an eye on things.
must be a personality thing. (that some have a problem...some don't). We don't do bones very often (chews more often) but in either case, one each, they take them, put them down, investigate each other's, top dog gets first pick, (but no real fighting or anything) and then they settle down and chew (until they get tired of it, and "bury" it under their crate pads.
Each of them has a corner/end of the crate pad/bed that is a couple of inches higher because of the treasures hidden underneath.