Looking at 1 Corinthians 6
“Do you not judge those who are within the assembly? Does any one of you, when he has a case against his neighbor, dare to go to law before the unrighteous and not before the saints? Or do you not know that the saints will judge the world? If the world is judged by you, are you not competent to judge trivial cases? Do you not know that we will judge angels? How much more matters of this life! So if you have law courts dealing with matters of this life, do you appoint them as judges who are not esteemed in in the assembly?
I say this so you will turn in upon yourselves and look for some wise man who will be able to make decisions between the brethren. But instead, one brother takes another to court—and does so in front of unbelievers! The very fact that you have lawsuits among you means you have been completely defeated already. Why not rather be wronged? Why not rather be cheated? Instead, you yourselves cheat and do wrong, and you do this to your brothers and sisters. Do you not know that wrongdoers will not inherit the kingdom of God? Do not be deceived because that is what some of you were. But you were washed, you were sanctified, you were made righteous in the name of the Lord Jesus Christ and by the Spirit of our God.
All things are lawful for me, but not all things bring people together. All things are lawful for me, but I will not be brought under the power of anything.”
In context of Christians judging matters between themselves Paul ends his argument here. After this he goes on to talk about why immorality is bad & somewhat hearkens back to the idea of why it is bad to have an immoral person judge your disputes. Paul says "All things are lawful for me, but not all things bring people together. All things are lawful for me, but I will not be brought under the power of anything." Most translations say, "not all things are profitable" or "not all things are beneficial." While there is truth there, another meaning of this conveys "not all things bring people together." Paul says that, while there is no law for Christians what we do still affects other people. Not everything we do brings us closer to one another, thus there were disputes among the people. Without laws or wise judges they could not decide how to settle matters in their community. Paul is pointing back to what he said about them finding a wise man to judge in their community. The judge's purpose would not be to "bring people under his power" or "exercise authority" over the brothers" but to bring people together.
"Be brought under the power" is the same word Jesus uses in Luke 22:26 for "have authority over." He said, "The kings of the Gentiles lord it over them; and those who have authority over them are called 'Benefactors.' But not so with you." Jesus says that the family of God will not "have authority over" one another & that they should not attempt to "bring others under false authority." Paul echoes the same thing basically saying, "Though I want you to identify judges to handle problems in the community those judges are not to have authority over you." It deeply relates to Romans 13 where He says to "voluntarily cooperate" with these judges. Seeing that the children of God are honored, Paul speaks of an "honor system" in regards to having wise judges to settle disputes, bring people together, & dole out punishments. It is an honor system because the people aren't required to have these judges nor are they utterly forced to cooperate with them. The judge cannot force punishment but the community can bring someone to him to have that person judged/punished. This is a similar system to what God set up in the Old Testament before there were kings. But in the assembly it would not operate by law, instead it would operate by the proven wisdom coming from the Spirit.
Looking at Romans 13
“Do not be overcome by evil, but overcome evil with good. All people are to voluntarily cooperate with excellent authorities. For there is no authority except under God, and those excellent authorities which exist are established under God. Therefore whoever resists such authority resists the order of God, and those who resist will bring judgment on themselves by the community. For judges are not a cause of fear for good behavior, but for evil. Do you desire not to fear authority? Do good and you will receive its commendation. Excellent authority is of God, it is a servant for your good. But if you do wrong, be in fear, for it does not bear the sword in vain. For of God it is a servant who punishes the wrongdoer. Therefore it is necessary to be in voluntary cooperation, not only because of punishment, but also for conscience sake. This is why you pay tribute tax because the judges are servants of God, devoting themselves to administering punishment on the wrongdoer. Give to all what is due them: tribute tax to whom tribute tax is due; toll to whom toll; respect to whom respect; honor to whom honor.”
Overall Paul is referencing a communal judge system to handle disputes. He talks about the same thing in 1 Corinthians 6 where He wishes the Corinthians would judge matters amongst themselves rather than going into secular court. The difference is that these judges already existed in Rome. Paul is making a case for why people should submit to excellent authorities (judges) that are are in the assembly in Rome. Paul speaks of community selected judges who naturally arise in the community because of their wisdom, who's authority is not found in themselves. Their authority is either supported by the community or by voluntary cooperation.
This isn't a model people should try to copy & force into existence. This can only occur organically in close Christian communities. In the assembly at Rome this had already happened so Paul addressed the people regarding the judges they had telling them to only submit to excelling judges. Excelling because they continue to handle cases in wisdom, love, & justice. They are seen by how they have judged their most recent cases. Paul made this distinction of excellence because apparently the Romans had some bad judges who were not worth submitting to. Likewise the Corinthians had appointed immoral people, who Paul identified as "so-called brothers" in 1 Corinthians 5:11, to judge their matters.
Also mentioned here is the fact the people paid “tribute taxes" to the judges. The word “taxes” is a little misleading as it implies a compulsory payment that is decided by the government receiving. A tax must be paid, regardless of whether anything has been received in return. Paul is instead talking about a voluntary payment, not a compulsory levy. A tribute tax is a specific ”tax” that is a symbol of submission and dependence. When judges do their work well, the whole of society benefits, not just the people who get their cases heard. He is telling Christians that they should decide what they owe to the judges in their community and make sure they pay them something.
Paul lets the people know who the trustworthy & excellent judges in the community are by saying they
Are under God
Established by God (organically by the Spirit over time)
Do not cause fear
Do not do terrible things to good people
Commend people for doing good
Exist for your good
Are servants of God (Christians)
Contrast that list with truths about secular governments & it is obvious that this passage cannot be referring to them.
Not Legal Judges
These judges weren't law enforcers. They operated relationally to settle disputes between people & to punish people the community deemed dangerous. "Judges hold no terror for those who do right, but for those who do wrong." This statement is only true if there were no laws. If there were laws then the judges would have authority outside of voluntary or communal court settings. They would be a terror to the people by trying to force them to do good. These judges did not have the responsibility or authority to force people to lead virtuous lives. They could only restrain harm by punishing crimes & reconciling people. They weren't to attempt to make people be on their best behavior. Again "all things are lawful" & "no one will have authority over you" but not all things bring people together, not all things benefit so there are times where it is wise to submit ourselves to other people's help & guidance. No one person is to manipulate us or force us to do anything but a lack of independent authority in God's family does not exempt us from the usefulness of cooperation.
For an in depth look at judges in the Assembly of the Old & New Testament read God Raises Good Judges
For an in depth look at Romans 13 in regards to judges read Understanding Romans 13
Judges in the Assembly Part 1: Why Romans 13 is not about Secular Government
The Letter of the Law, The Life of Love -1 Peter 2:11-17: Focuses on scripture that does address how Christians relate to secular government.
Did Paul Shame People?: Most translations of 1 Corinthians 6 use the word "shame." See why this is inaccurate.