As strange as it sounds, this is the sort of thing which happened in the Roman world. Dio Chrysostom reports that the Roman word of the late first century was filled with “lawyers innumerable, twisting judgments.” (Cited by Winter, After Paul Left Corinth, 62). These lawsuits were politically motivated, between members of the rich and elite class (or want-to-be elite.) These lawsuits were opportunity for young orators to show off their rhetorical talents before the elite citizens (the judge, magistrate, jurors, etc.)
Paul’s solution to the problem is to “shame” them for suing their brothers. Shame is an important factor in first century personal politics. Paul says twice in this letter that he desires to put the church to shame over some behaviors (here and drunkenness in chapter 11.) If the lawsuits were motivated by a perceived loss of honor in the first place, Paul turns a popular expectation upside down by saying that it is a loss of honor for a Christian to take his brother to court.
This therefore is the “shame”: they are suing family members. Paul frequently refers to his readers as “brothers” to emphasize that the Church is a new family rather than a social club. A person is not suing some stranger who has insulted them, they are suing brothers. The Romans did not approve of intra-family lawsuits, therefore Paul is emphasizing brotherhood of the believers.
Paul does not recommend going through a private arbitrator to solve disputes, as was the right of citizens. He says that they church ought to be able to deal with such disputes within the family. There are people within the congregation, presumably, that are styling themselves as orators, and all of the citizens would be familiar with the process of arbitration. Paul is saying that the church ought to function like a family, brothers dealing with one another with “strife and discord.”
In fact, of one were really living out the teaching of Jesus there would be no need to sue over a perceived insult. The brother forgives his brother. Given the cultural background above, this is counter-cultural and radical!
How do we “bridge the gap” and apply this sort of teaching in a modern, local church context? At the very least, the church needs to return to the truth than all members of the Body of Christ are brothers and that it is a loss of honor to treat a family member like a stranger. This alone would have a positive effect on the local church.