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Each church in the 1st century was locally governed (unlike the Catholics today).

If you mean by locally governed that there was someone in charge locally to handle matters, yes that's true. This person was the episcopos, whence we get our word "bishop". But if you mean that this bishop was out on his lonesome and that there wasn't any oversight from anywhere else, you may want to read Clement's Letter to the Corinthians, which was written at the tail end of the 1st century at about A.D. 96 or so.

Clement was the bishop of Rome in the 90s A.D. and he writes to the Corinthians as follows:

The Church of God which sojourns at Rome, to the Church of God sojourning at Corinth, to those who are called and sanctified by the will of God, through our Lord Jesus Christ: Grace to you, and peace, from Almighty God through Jesus Christ, be multiplied.
Owing, dear brethren, to the sudden and successive calamitous events which have happened to ourselves, we feel that we have been somewhat tardy in turning our attention to the points respecting which you consulted us

Here we have a bishop of Rome, whom Eusebius calls the 3rd successor to St. Peter, involving himself in a dispute at Corinth. Moreover, the Corinthians seemed to have asked him to get involved in this local dispute. Clement answers by taking a group of schismatics at Corinth to task for trying to subvert the legitimate heirarchy that is there and tells them to respect the establishment there in humility.

Also, Clement suggests that, far from a mere collection of a few believers, the Church in Corinth is a visible, fixed entity with an appointed hierarchy who were passed down from the Apostles. See especially his Chapters 42 and 44:

The apostles have preached the Gospel to us from the Lord Jesus Christ; Jesus Christ [has done so] from God. Christ therefore was sent forth by God, and the apostles by Christ. Both these appointments, then, were made in an orderly way, according to the will of God. Having therefore received their orders, and being fully assured by the resurrection of our Lord Jesus Christ, and established in the word of God, with full assurance of the Holy Ghost, they went forth proclaiming that the kingdom of God was at hand. And thus preaching through countries and cities, they appointed the first-fruits [of their labours], having first proved them by the Spirit, to be bishops and deacons of those who should afterwards believe. Nor was this any new thing, since indeed many ages before it was written concerning bishops and deacons. For thus says the Scripture a certain place, "I will appoint their bishops s in righteousness, and their deacons in faith."...
Our apostles also knew, through our Lord Jesus Christ, and there would be strife on account of the office of the episcopate. For this reason, therefore, inasmuch as they had obtained a perfect fore-knowledge of this, they appointed those [ministers] already mentioned, and afterwards gave instructions, that when these should fall asleep, other approved men should succeed them in their ministry. We are of opinion, therefore, that those appointed by them, or afterwards by other eminent men, with the consent of the whole Church, and who have blame-lessly served the flock of Christ in a humble, peaceable, and disinterested spirit, and have for a long time possessed the good opinion of all, cannot be justly dismissed from the ministry. For our sin will not be small, if we eject from the episcopate those who have blamelessly and holily fulfilled its duties. Blessed are those presbyters who, having finished their course before now, have obtained a fruitful and perfect departure [from this world]; for they have no fear lest any one deprive them of the place now appointed them. But we see that you have removed some men of excellent behaviour from the ministry, which they fulfilled blamelessly and with honour.