In the text, we hear Shagrat and Gorbag griping about their bosses, and reminiscing about old times. A little later, we learn that shortly after this conversation, Shagrat killed his old buddy Gorbag for trying to steal Frodo's stuff, and that is the last we hear about either of them.
According to the wikis, however, Shagrat took Frodo's mithril shirt and some other possessions to Barad-Dur, turned them over to Sauron, and was immediately killed for letting the prisoner escape. This would obviously explain how the Mouth of Sauron obtained these items, which he later presented to the Captains of the West in front of the Black Gate.
But I can't find any direct quotes from Tolkien himself on this subject. It certainly isn't surprising that Sauron, who isn't a very nice guy even on his best day, would kill his servants for allowing a supposed "spy" to escape. In fact, it almost seems inevitable that Sauron would have killed him for his perceived failure. But I want to know what Tolkien himself had to say about it.
So did Tolkien himself ever discuss Shagrat's ultimate fate?