The scripture passage in question reads: "If I say to the wicked, 'You shall surely die,' and you give them no warning, or speak to warn the wicked from their wicked way, in order to save their life, those wicked persons shall die for their iniquity; but their blood I will require at your hand. But if you warn the wicked, and they do not turn from their wickedness, or from their wicked way, they shall die for their iniquity; but you will have saved your life" (Ezekiel 3.18-19). Removed from its fuller context it certainly seems that it is up to the individual to point out the sins of others. However, in context it is clear God is speaking specifically to Ezekiel and the warning cannot be applied to the general reader. Indeed, if one chooses to accept this verse as a legitimate command for themselves, they would also have to accept the odd eating and drinking commands God gave Ezekiel, including baking bread on dung in the street where everyone can watch (Ezekiel 4.12). Yuck!
On the other hand, there are passages in the New Testament where confronting sin is expected. In James we read, "My brothers and sisters, if anyone among you wanders from the truth and is brought back by another, you should know that whoever brings back a sinner from wandering will save the sinner's soul from death and will cover a multitude of sins" (James 5.19-20). Clearly this indicates sin is to be confronted, acknowledged, and dealt with. However, the motivation for confrontation in this passage is one of love, not condemnation.
Indeed, those who set themselves up in the judging business are in clear danger. Jesus himself said, "Do not judge, so that you may not be judged. For with the judgment you make you will be judged, and the measure you give will be the measure you get. Why do you see the speck in your neighbor's eye, but do not notice the log in your own eye? Or how can you say to your neighbor, 'Let me take the speck out of your eye,' while the log is in your own eye? You hypocrite, first take the log out of your own eye, and then you will see clearly to take the speck out of your neighbor's eye" (Matthew 7.1-5). Jesus' point is this-even if your neighbor has a BIG sin in their life, if you're judging, you sin is even bigger. Period.
Now, some will say that Matthew 18 provides guidance on how to confront another's sin. And since the instructions are there, the implication is we should be addressing other's sins. However, Jesus is very specific that the confrontation is only allowed when someone "sins against you" (Matthew 18.15 italics added). If someone sins against another, or if the sin is against God, then, in the words of Hebrews 10.30, "'Vengeance is mine, I will repay.' And again, 'The Lord will judge his people.'"
On the other hand, Paul wrote to the Corinthians, "But now I am writing to you not to associate with anyone who bears the name of brother or sister who is sexually immoral or greedy, or is an idolater, reviler, drunkard, or robber. Do not even eat with such a one" (1 Corinthians 5.11). Certainly there seems to be an element of judging going on here, but note the element of confrontation is notably absent. Instead, in an effort to keep oneself from falling into the sin and addictions of another, Christians are encourages simply to avoid other so-called Christians who have stumbled on the path.
But lest some think this gives them to right to drive these people out, Jesus told this parable: "The kingdom of heaven may be compared to someone who sowed good seed in his field; but while everybody was asleep, an enemy came and sowed weeds among the wheat, and then went away. So when the plants came up and bore grain, the weeds appeared as well. . . . The slaves said to the master, 'Do you want us to go and gather the weeds?' But the master replied, 'No; for in gathering the weeds you would uproot the wheat along with them'" (Matthew 13:24-29, selected).
So, are Christians supposed to be busy pointing out other people's sins? Although James did write that helping another from their sins was a good thing, he also was very clear that mercy triumphs over judgement (James 2.13). Berating, condemning, and judging others is specifically prohibited in the Bible and those who judge are in danger themselves of even a more severe judgement.