The passuk statesהוכח תוכיח את עמיתך give rebuke to your friend. The passuk ends ולא תשא עליו חטא and you will not receive for it sin. Which rebuke does the Torah refer to and what is the sin?
The Ramban explains when a person does bad to you do not hate him, rather rebuke him, let him know you are disturbed. This may cause him to explain himself or repent for his improper doing. The passuk follows ולא תשא עליו חטא if you let him know you are upset, you will not transgress the issur mentioned at the beginning of the passuk, לא תשנא את אחיך בלבבך, do not hate your brother in your heart, since you’ve notified him about your issue.
Rashi explains differently; הוכח תוכיח do not rebuke your friend in public which will embarrass him, for doing so will cause you to carry the sin of his embarrassment. The first part of the passuk teaches that there is a mitzvah to rebuke, and the second part clarifies the appropriate way to give the rebuke; not in public rather in private. The Gemara derives from the double lashon הוכח תוכיח to rebuke even a hundred times, if one repeats sinning. And ולא תשא עליו חטא teaches do not rebuke if it will cause his face to change. Rashi explains this refers to one who rebukes in public and causes the persons face to change to a white color, embarrassing him.
Onkelos translates the words ולא תשא עליו חטא and you will not receive punishment for his sin. The Ramban explains this means if you do not give rebuke you will receive the punishment of the sin which he transgressed, it is as though you did it. The reason is because had you given rebuke, he would have stopped and not performed the sin. Since you did not give rebuke you enabled and allowed for the sin to be done. Therefore you will be punished as though it is your sin.
The Mishnah teaches that Rebi Eiezer Ben Azariah had a cow which went out on Shabbos with a strap between its horns against the interest of the Chachamim, since they considered this carrying a package on Shabbos. The Beraisa explains that in truth this wasn’t his cow, rather his neighbor’s cow. However since he did not protest, it is considered his animal. So too anyone who is able to protest about wrong doing in his house and doesn’t, he will be included in the punishment of his household; in his city, he will be included in the city; in the entire world, he will be included in the world.
The Iyun Yaakov says that from the bad we can derive to the good which is always greater. If a person helps the people of his house to do good, he will be rewarded with the mitzvah of his household, and so too is with his city and the entire world. The Midrash brings that if one helps his household to perform good, the mitzvah will bring him to do more and more good, being able to influence his city and then the entire world. However if he does not influence his household, the bad will be enhanced, causing his city to be influenced and then the world.
May Hashem help us rebuke in the appropriate way and influence people to do mitzvos.