We are not worthy of this title on our own merits. God only chose us because we are the descendants of the first ethical monotheist, Abraham (Genesis 18:19). That's the sole reason. In fact, to prevent ourselves from any arrogance that might be compelled by this title, one has to look no further than episodes throughout Tanach, where we have often failed to live up to God's standards. This is hardly a tradition based on inherent superiority. Instead, goodness is always based upon a person's overall behavior.
Two of the clearest examples which demonstrate that chosenness has nothing to do with innate superiority are the stories of Noah and Ruth. Noah was not Jewish, but was saved from annihilation because "he was the most righteous person in his generation" (Genesis 6:9). Similarly, Ruth was not born a Jew, but due to her righteousness was rewarded with having the Messiah himself, Mashiach Ben David, descend from her. Once again, good deeds are more important than who your parents happened to be.
While we have the responsibility of being "a light unto the nations" (Isaiah 42:6), we first have to be a light unto each other. As the saying goes, "charity begins at home." Still, we can't let this task overwhelm us to the point where we are either only good to Jews or only good to non-Jews. The more consistently we try treat other people well, the easier it will become to achieve both outcomes. Whether we're in shul on Friday night or out shopping during the week, we must always strive for top-notch behavior. Because of who we are, the world scrutinizes everything we do. At times, this can be frustrating, but it isn't always a bad thing. It can also be a great opportunity to show people the kind of character God truly desires.
Other peoples and religions have tried to strip us of the Chosen People role, but it's doubtful they actually realize what such a label entails. It's not all it's cracked up to be, and it's not something that can be transferred. As a prominent non-Jew, the Reverend Edward H. Flannery, put it: "It was Judaism that brought the concept of a God-given universal moral law into the world . . . the Jew carries the burden of God in history [and] for this has never been forgiven."
Due to the antagonism and responsibility this title brings, there's an old joke that most Jews would have preferred if God had chosen someone else. Nevertheless, spreading ethical monotheism to the world rests on our shoulders. The Chosen People idea is not dogma but historical fact. It's a powerful concept, but nothing to brag about. It's simply a calling we must try to live up to.