Yes, Jesus does actually talk about what to do when another person sins.

 Matthew 18. A rarely quoted passage these days. Specifically verses 15-17, but the whole context is important. Here’s what Jesus says: 

“If your brother sins (some have – against you), go and point out his fault, just between the two of you. If he listens to you, you have won your brother over. But if he will not listen, take one or two others along, so that every matter may be established by the testimony of two or three witnesses. If he refuses to listen to them, tell it to the church. And if he refuses to listen even to the church, treat him as you would a pagan or a tax collector.” (emphasis mine)
Let’s point out some important things in these few verses. 
First, when a fellow believer sins you are to go and point out the sin. Yes, you are to make a call on someone else’s actions. You cannot know the heart, but you can see the actions and when someone is definitely wrong (i.e. in sin) it’s ok and Biblical to acknowledge it.
Second, Jesus states “IF he listens to you, you have won your brother over…” I’d like to what ask from what? From what have you won them over? Maybe letting the sin get worse and bringing greater discipline or even punishment upon themselves. Maybe something else. Either way, it’s seen as a good thing for the person who sinned if they listen and STOP.
Third, if the believer refuses to listen even to the church what do you do? It sounds harsh and completely unloving to us today, but you are to turn that person out of the fellowship of the church. You hardcore discipline them bc that’s how important sin and repentance are to God. 
Where’s the “do not judge” message here? Treating the person as a ‘pagan’ or ‘tax collector’ essentially meant – don’t associate with them.
Obviously not all, but much of the Jewish community at this time were zealous in their desire to obey God and keep His commands (tax collectors were not known their obedience). So zealous were many Jews, that often they took it to the extreme of not being in company with those who did compromise (such as tax collectors). You didn’t invite them to dinner, you didn’t talk to them on the street, and you only associated with them if you absolutely had to. And as for pagans… it was rare for an obedient Jewish person to closely associate with one, unless necessary.
(There are reasons for this but I don’t have to time to get into the background issues)
But (you may say) Jesus associated with tax collectors and sinners! Matthew, one of the disciples was a tax collector! True. But what happened to them after Jesus ate with them?? Look at Zacchaeus or Matthew for instance. And did He associate with every tax collector and sinner He saw? Nope. There were many who didn’t receive any attention from Him, and those who did were changed. The tax collectors and sinners Jesus associated with were wanting His company and His teaching, to hear and obey this amazing Rabbi. 
Jesus didn’t associate with ‘sinners’ bc they were sinners. He intimately dealt with those who were desiring to repent and change. How much time did Jesus give Herod (a definite sinner)? Or what about all those pagan towns who could have used some godly teaching and love? Why didn’t Jesus spend more time there?
And truthfully, if Jesus saying ‘treat them as a pagan or tax collector’ meant simply “forgive and forget” then why even bother going through the rest of it? Why bring in other people? Why take it to the church if you’re just going to end up ignoring the issue? 
So what’s the point of this harsh discipline? The same as it is behind any discipline. Restoration. The point isn’t the punishment, but instead the hope of the restoration of the believer. 
You want to restore the person to an intimate relationship with God and a right relationship with you and other believers. I discipline my children not bc I’m angry but bc I want them to live rightly according to God’s standards. I want my children to have the best life possible, and I know that can only happen when they are living life God’s way by obeying His commands. 
Hard but real love happens when I discipline my children with that goal in mind. If I were to let my children continue sinning that would be easier but not more loving. What could happen to them? Jail… or worse.
“Do not judge” sounds loving and really nice. But honestly, it can be the most damming thing we can do to people. When a fellow believer is in sin, one of the most loving things we can do is pray for them, and lovingly gently, honestly speak to them about the sin. 
Yes, there are right and wrong ways to do this. Yes, there are people who have more right to speak into someone else’s lives than others. Yes, there are times and places for this and obviously wrong times and places.
But the point is – acknowledging sin isn’t judging. And actually, it can be the most loving thing for a person WHEN it’s done correctly. So, may we take back what Jesus said and apply it correctly. May we acknowledge our sin and openly admit we are not perfect. May we be willing to humbly learn from others. And may we love our fellow brothers and sisters in Jesus enough that we are willing to humbly confront them if God calls us to do this.
(If you want to see Paul’s take on sin in the church read the books of Corinthians, especially 1 Cor 5 and 2 Cor 7. It may sound a bit familiar now)