How should we respond to each other’s sin? 
Five commands in Scripture for dealing with sin in another person.

1. Teach one another (didasko, “to teach”) 

God requires that you let the Word dwell in you richly enough that you can teach and admonish in all wisdom (Col. 3:16). The advice we give each other should come from Scripture. 

2. Expose sin (elegko “to show a fault”) 

Matthew 18:15 “If your brother sins against you, go and show him his fault (elegko), just between the two of you. If he listens to you, you have won your brother over.”

This involves both exposing the wrongness of the action and persuading the person of guilt (conviction). That dual purpose is seen repeatedly in Scripture where the word elegko is used:

Expose the wrongness of an action:

Ephesians 5:11,13 Have nothing to do with the fruitless deeds of darkness, but rather expose them. … everything exposed by the light becomes visible….
1 Timothy 5:20 Those who sin are to be rebuked publicly, so that the others may take warning.
Persuade the person of his guilt:
1 Corinthians 14:24 But if an unbeliever or someone who does not understand comes in while everybody is prophesying, he will be convinced by all that he is a sinner and will be judged by all….
2 Timothy 4:2 Preach the Word; be prepared in season and out of season; correct, rebuke and encourage—with great patience and careful instruction.
 Titus 1:9,13 He must hold firmly to the trustworthy message as it has been taught, so that he can encourage others by sound doctrine and refute those who oppose it. … This testimony is true. Therefore, rebuke them sharply, so that they will be sound in the faith. …
Once the sin is exposed, most Christians will respond with sorrow and repentance. At that point, you can help restore a person. But if someone does not repent, he needs rebuke.

3. Rebuke one another (epitimao “rebuke, censure, warn, or punish”) 

Luke 17:3 “So watch yourselves. If your brother sins, rebuke him.” 
This term is aimed more at the will than the mind. It goes beyond persuading the person he has done wrong; it has the goal of persuading someone to change. A rebuke is not a suggestion; it is a command that calls for action:

Matthew 8:26 Then he got up and rebuked the winds and the waves, and it was completely calm. 

4. Admonish one another (noutheteo “to admonish, to instruct or warn in an effort to correct behavior”) This word is behind the phrase “nouthetic counseling.” 

1 Thessalonians 5:14 And we urge you, brothers, warn (lit., admonish) those who are idle, encourage the timid, help the weak, be patient with everyone.
2 Thessalonans 3:14,15 If anyone does not obey our instruction … warn (lit., admonish) him as a brother.
Admonishing is showing a person his behavior and comparing it with what the Bible says.
Colossians 3:16 Let the word of Christ dwell in you richly as you teach and admonish one another with all wisdom.…
Admonishing is not the same as shaming—but it is similar enough that it seems a lot like shaming. That’s why Paul had to clarify what he was doing in admonishing the Corinthians:
1 Corinthians 4:14 I am not writing this to shame you, but to warn (lit., admonish) you, as my dear children.
  The word translated shame means to cause someone to look down on himself. We should never do that, but true admonition often looks very much like that. It is tough love. 
Sometimes people become so desensitized to the convicting ministry of the Holy Spirit that gentle words simply do not get their attention. That is when they need you to come along and give them a kick in the seat of the pants. You should have the courage and strength to do that, and the wisdom and love to know the difference between admonishing and shaming. 
Sometimes the Greek word noutheteo is translated “warn” rather than “admonish.” Admonition involves warning a person about the consequences he is in for if he does not change. Warning is helpful for someone ensnared by sin. Sin causes a person to lose perspective. The pleasures of sin are only for the moment; they can never compare with the rewards of righteousness in the end. When you warn someone, you point out the long-term consequences, making it easier for him to see his foolishness.

5. Provoke one another to love and good deeds (paroxusmos “provoke, incite”) 

Like the English word “provoke,” the Greek word paroxusmos is normally used negatively. 

1 Corinthians 13:5 … [love] is not easily angered … (lit., love is not provoked to anger).
Acts 17:16 Now while Paul was waiting for them at Athens, his spirit was being provoked (paraxuno) within him as he was observing the city full of idols.
When used in a negative way, the word “provoke” means to do something that incites a response of anger. The writer of Hebrews tells us to do things that incite people to love and good deeds:
Hebrews 10:24-25let us consider how we may spur (lit., provoke) one another on toward love and good deeds. Let us not give up meeting together, as some are in the habit of doing, but let us encourage one another—and all the more as you see the Day approaching. 
When you are counseling someone who is in sin, after exposing his fault to him, rebuking him, admonishing him, and warning him, you must then seek to provoke him—not to anger, but to love and good deeds. Inflame the person—not to wrath, but to the Christian way of life.
It is important to understand that this step comes only after there has been genuine repentance. The goal of rebuke and admonition is to bring repentance. If there is no repentance, encouraging a change in behavior is a waste of time. 
For example, a person with a habitual sin may come to you for counsel because he is suffering painful consequences for his sin. He wants the consequences to stop, and so he wants you to help him modify his behavior. But if he is not repentant over the sin in his heart, merely helping him modify his behavior may do more harm than good. If the painful consequences of sin are removed, the motivation to repent is lessened. And if the behavior is altered to his satisfaction but he never repents, he will continue to dishonor God and do harm to himself spiritually.
Once a person has repented, helping him make a change in behavior is appropriate. This is where provoking one another to good deeds comes in. When you provoke someone to anger, you “push his buttons” by doing things that are sure to get a reaction from him. Provoking someone to good deeds is the same way. You find out which buttons you can push to stimulate that person toward godliness. This requires knowledge of the individual as well as creative insight.