When the Chofetz Chaim said that if a person consistently is transgressing a prohibition between man and G-d which is known to everyone to be against Jewish law and he is doing it on purpose this means it has become part of his way of life, not a sin that he’s done once or twice. If you only saw him do it once or twice then you have to assume maybe he repented but if this is something which is known as part of his lifestyle and he is purposely doing it and it6a sin which everyone knows is a problem, for example every Jew, even those non-observant know that shellfish, pig, and milk and meat together are against Jewish Law so if a person is known to go out to eat at McDonalds and order a bacon cheeseburger all the time you are allowed to speak out against him as long as you meet the 7 prerequisites and speak out in public in front of at least 3 people so that word will spread and no one will suspect him of flattery. One way a sin perpetrated against his fellow man like stealing or hurting someone is different than a sin between man and Hashem is that for the most part besides maybe stealing, if a person is bullying someone physically, emotionally, or monetarily he usually does it because he thinks the victim deserves it. Many times, he has excuses and justifications why he is doing what he is doing he is not outright sinning maliciously just to spite Hashem therefore the reason why you are allowed to speak out against him is for the sake of justice to protect the innocent and the victims and resolve the problems the perpetrator caused if he hasn’t already fixed the issues himself. The perpetrator is still considered “part of your nation” and all the mitzvos that apply to a fellow Jew still apply to him like “Love your neighbor as yourself ” or “Don’t hate your brother in your heart”. However, a person who is habitually transgressing a sin between man and Hashem on purpose which everyone knows is wrong like eating bacon cheeseburgers then he has left the fold and is not considered part of “your nation”. For this reason, one is allowed to speak lashon hara about home, insult him and does not even have a mitzvah to rebuke him since the Torah says by these three mitzvos that who you speak to must be considered part of “your nation”. If a person decides to purposefully leave the fold and not adhere to what everyone knows is wrong, then he does not have to be treated the same way as everyone else. By speaking out against and speaking down at him that might pressure him to repent. According to Jewish law you don’t even have to rebuke him directly because at this stage of the game he has the same status as a scoffed who you can assume does not listen to rebuke but it is still a mitzvah and a nice thing to do to try to engage him personally and convince him to change his ways and maybe you wouldn’t have to speak out against him to apply pressure, but there is no obligation to rebuke such a person.