The Scriptures teach local congregational independence and autonomy. What does this mean?
The body of Christ (universal terminology as in one body, Ephesians 4:1-3) is not some interconnected web of congregations; rather, the body is composed of individuals (see I Corinthians 12:12-27). Individual Christians ought to seek out other Christians in their locale with whom they can work, worship, and edify one another. This is what Christians have done from the beginning (Acts 2:42). Fellowship requires togetherness.
Local congregations are made up of Christians who purposefully join together to that end (e.g., "to the church of God that is in Corinth," I Corinthians 1:2). A local congregation is not institutionally tied to any other congregation, but stands on its own as an independent, self-governing group. One local church is not amenable to another local church; they each stand independently before Christ, accountable to Him. Revelation 2-3 gives us a good snapshot of this. The Lord addressed each congregation on its own, gave the good and bad where each existed, and told them what they needed to be doing. The church at Ephesus had no authority over the church at Smyrna, and vice versa. That is the position of Christ, the King of all.
To clarify further, saying that a church is autonomous or self-governing does not imply that the group is not under Christ as Head or that such a group can just make up its own rules about what it does. In this context, autonomous simply means that a local group takes care of its own business without being controlled or overseen by any other group or institution. When a group has shepherds, the shepherds are to oversee only that group which is among them (Acts 20:28; I Peter 5:2). They have no biblical warrant for taking oversight of other congregations or even asking other congregations to bring their work under them. In the first century, the apostles operated on a level larger than local churches because of their special roles chosen by Christ, but that is not so for elders or shepherds. Absent living apostles now, each biblical congregation would seek to abide by what the apostles and prophets left behind, but no living person today has any authority over any group of churches.
Practically, what this means is that each congregation makes decisions that the leaders believe will work best for that group alone. Given that all should be seeking to act within the principle of abiding in God's word, the decisions for one group will not necessarily be the best for another. One group may decide that meeting once on a Sunday is best for them; another may decide to meet twice at times that vary from other groups. One group might use a sign simply indicating that Christians gather there, while another might use a different description (assuming all are biblical). It is not the business of anyone outside the local group to become the judge of that group (e.g., "that church quit meeting on Sunday night; they must be slipping in their spiritual focus"). When decisions are made within the realm of what Scripture allows, there is a good bit of leeway and many choices to be made within that group's purview. Those who are not part of that group have no warrant to interject themselves into the group's business, and we all need to be very careful in how we criticize decisions made by an autonomous group of which we are not a part. We are neither Christ nor the apostles.
It should be a given that all disciples of Christ should be seeking to abide within His word (John 8:31). None of this is to say that we cannot evaluate actions of a group on the basis of Scripture. However, we need to be careful that we are not misjudging what is a local congregational liberty in acting upon what best works for their situation. Many choices must be made, and local leaders are usually aware of their own member needs. How can we judge what we don't even really know? Even within a local congregation, the elders will likely be dealing with issues surrounding some members about which other members will not have all the information. How much more when we aren't even part of a group!
Respecting autonomy is important, for there is no super-organization that controls and commands what each group is to be doing. No one person or other outside group or institution has authority to take over a local church. That is the realm of the Lord and His word. Now we might assess a local situation for ourselves and decide that they are doing what we cannot participate in, or, on the flip side, that they are doing what is right and the situation fits us well. At the end of the day, that's pretty much the only way it can work. Individuals must decide. Local groups must decide. Who are we to try to usurp the Lord's position?
Respect autonomy. The alternative is neither scriptural nor pretty.