Despite some publicity, including some famous photos in Life and Look magazines, this was not a common practice at all. Most skulls never left the battle area. Skulls that were taken, were posed and photographed under battle stress within the battle area usually as a type of intimidation macho, mostly for the enemy to see. Once the shooting stopped, few who actually did the bloodiest fighting were interested in actually taking them home. Those who did, tended to be rear-area souvenir hunter types who cruised the battle areas later, after the shooting had stopped and when it was safe to walk around. It was a court-martial offense if you were caught. Most of those who did try to smuggle them back, were indeed caught, as returning bags and boxes from combat zones are always checked by customs, logistics and safety officers to make sure live rounds, weapons and ammunition are not being brought back from the combat area, and most skulls were intercepted that way. The few skulls that did make it back usually had short lives. Few spouses wanted or tolerated such things around the house. Animals would smell them and dig or chew at them, and curious children or grandchildren might be around, and then there's a lot of embarrassing explaining to do. In addition, if the skull fell into the wrong hands or was observed by the wrong friend or neighbor, who wants to explain to the local police that it is simply a gruesome war trophy (which makes you look like a psycho) or that you are not the local serial killer.
So while no records were kept of such things, there's still a story or two floating around out there from old time combat veterans of burying them in a backyard or nearby patch of woods, mostly all within 5 years or less of coming home. Time tends to heal all wounds and the civilizing and maturing influence of continued peace tends to ease PTSD and change perspective.