They devoted themselves to the apostles' teaching and to the fellowship, to the breaking of bread and to prayer. Everyone was filled with awe, and many wonders and miraculous signs were done by the apostles. All the believers were together and had everything in common. Selling their possessions and goods, they gave to anyone as he had need. Every day they continued to meet together in the temple courts. They broke bread in their homes and ate together with glad and sincere hearts, praising God and enjoying the favor of all the people. And the Lord added to their number daily those who were being saved. ~ Acts 2:42-47
That's a fine place to begin ... but to ignore the rest of the chapter (except, perhaps, a few verses to prove a point about baptism) is to excise the reason why this new church - at least for a time - worked to God's glory:
The Holy Spirit.
He is present in a powerful way throughout Acts 2.
I believe it is impossible to have a vibrant, growing, scripturally-founded, God-praising, unbelieving people-pleasing, kingdom-representing church without the Holy Spirit.
And, in large measure, many churches and individuals have tried. Some actively rebel against the idea of a present, powerful Spirit doing God's work within and among them. Others downplay the possibility, or fear that abuses of the Spirit's gifts (more likely, the pretension of them) would cause too much trouble. Others simply don't seem to be aware of the full measure of the promise of Jesus regarding Him.
We like to quote Acts 2:38a: "Peter replied, "Repent and be baptized, every one of you, in the name of Jesus Christ for the forgiveness of your sins."
But we don't know what to do with Acts 2:38b-39: "And you will receive the gift of the Holy Spirit. The promise is for you and your children and for all who are far off—for all whom the Lord our God will call."
An Acts 2 church has no such confusion or fear or rebellion. An Acts 2 church believes what Jesus said, what Peter repeated through the Holy Spirit. An Acts 2 church lives their Christ-life in full view of others, lovingly, 24 hours a day, seven days a week, sticking together, at the religious meeting-house/temple and at home, breaks bread and fellowships others, adheres to the apostles' teachings, has everything in common, sells possessions and goods and gives to the poor.
What was their structure?
Jesus was their living King. He spoke to and through them in empowering ways via His promised Holy Spirit. The apostles - those closest to Him in the years that His mortal heart beat for them - shared His teachings.
"Yes," you might say, "... but who was in charge?"
Well, the answer is above.
"Yes," you might respond, "... but somebody has to be in control. Someone has to have authority."
Someone did: Jesus.
The apostles recognized that, acting with His authority through His Spirit, just as He had promised them, doing exactly what He had said they would do:
`Given to me was all authority in heaven and on earth; having gone, then, disciple all the nations, (baptizing them - to the name of the Father, and of the Son, and of the Holy Spirit, teaching them to observe all, whatever I did command you,) and lo, I am with you all the days - till the full end of the age.' ~ Matthew 28:18-20, Young's Literal Translation
Forgive me for not being a student of biblical languages nor a divinity-school graduate. I use YLT here to communicate the original grammar, structure and intent of the Greek text since I can't read it. What Jesus says here is not so much command as it is prophecy; acknowledgement in advance of what His followers will do with the authority He is sharing with them.
They knew He was in charge. It wasn't just a "given," it was the uncontested reality of the situation. There were no alternatives.
Our contemporary requirements to exercise local church autonomy or to submit to an overarching hierarchical human authority; to have church leaders - those are a comfort to us in the absence of our recognition of the reality of the century-one church.
Jesus' authority - given by God - is present in apostolic teachings and through the presence of His Holy Spirit. It's not an either-or. It's a both-and.
We have the necessary teachings preserved in scripture.
But if that's all we have, we're trying to do a two-handed job with one arm.
God wants to help us help Him, whether in daily living or church governance or any other matter. He offers Jesus' teachings and example and life and resurrection.
He also offers His Holy Spirit to assist, if we ask and are willing to receive Him.
An Acts 2 church knows that, and asks, and lives it.
One more time:
"If you then, though you are evil, know how to give good gifts to your children, how much more will your Father in heaven give the Holy Spirit to those who ask him!"~ Jesus, Luke 11:13