In Chapter 16, Bilbo sneaks out of the besieged mountain, offering the Arkenstone as a bargaining chip for the Elvenking to coax Thorin out before all the dwarves (and Bilbo) starve to death. When the Elvenking questions Bilbo, the hobbit describes himself with an oxymoron:
“But how is it yours to give?” [the Elvenking] asked at last with an effort.
“O well!” said the hobbit uncomfortably. “It isn’t exactly; but, well, I am willing to let it stand against all my claim, don’t you know. I may be a burglar—or so they say: personally I never really felt like one—but I am an honest one, I hope, more or less. Anyway I am going back now, and the dwarves can do what they like to me. I hope you will find it useful.”
The idea that Bilbo is an “honest burglar” is contradictory, but it also perfectly explains the person he has become over the course of his adventure. To answer the Elvenking’s question literally, the Arkenstone became Bilbo’s to give when he snuck into Smaug’s lair and squirreled it away, concealing from the dwarves that he had found anything special. All his run-ins with trolls, goblins, Gollum, spiders, elves, and more have trained him for this moment. Bilbo is not a blundering hobbit who has happened upon a treasure. By this point, he is a skilled and experienced burglar who has stolen one of the most valuable items any of the characters has ever seen. In a way, he has earned the Arkenstone.
At the same time, Bilbo seems to be missing one skill that is important to a burglar: he cannot keep his mouth shut long enough to take his prize home for profit. He conceals the Arkenstone from the dwarves for a time, but he reveals it to the Elvenking once he realizes that it might help facilitate peace. The men and elves seem ready to cover up Bilbo's involvement in the deal, but Bilbo himself comes clean with Thorin once negotiations have begun. Bilbo accepts Thorin's fury and allows himself to be called a traitor. He does his best to make amends by offering to consider the Arkenstone his share of the treasure, already spent on diplomacy. Thorin accepts but still expels Bilbo from the adventuring party. Bilbo thus commits a masterful burglary, only to give up all the riches and companionship he has been working toward for the entire novel.
At the start of the novel, Bilbo rejected the idea that he was a burglar. Gandalf billed him as one to convince the dwarves that he would be a useful member of their party. Bilbo did not feel he had the skill to be a burglar, and he did not like to think of himself as a dishonest thief. In the end, Bilbo manages the impossible: he proves Gandalf right without betraying his own ideals.