Summary: What is entailed in appointing elders? What rights and authority is conferred in the appointment process?
“When they had appointed elders for them in every church, with prayer and fasting they committed them to the Lord in whom they had believed.” 
The New Testament model for a church of the Lord Jesus differs significantly from what has been commonly accepted among the churches of our Lord’s Zion during the past century. The model of church governance that is provided in the New Testament is what we commonly refer to as congregational polity. Congregational polity should not be interpreted to mean that a church is a democracy—it is not. A mere majority does not of itself ensure biblical certitude. Congregational polity does mean that as each member submits to the reign of Christ among His people—He is the Head of the Church—the church is united in one spirit to discover the mind of the Master of the church and to obey His will. That holy mind and divine will is revealed through His written Word, which is to be received as inerrant and infallible. Thus, the Word is for a biblical congregation a perfect guide for faith and practise.
Yet another difference from contemporary models is that each church is autonomous. Though most Baptists give lip service to this truth, in practise it seems seldom to be followed. To say that a congregation is autonomous is simply a way of saying that each church stands as a separate entity. No outside agency is permitted to dictate to a church in matters of faith and practise. As a congregation, we may plead with other churches, admonish them and even declare them to be out of fellowship on the basis of deviant doctrine or because they permit moral/ethical contamination to continue unchecked; however, we have no authority over another congregation. In the same way, no other church or convention or fellowship or association or union has authority over our own congregation in matters of faith and practise.
Having stated that truth, I must hasten to add that churches are responsible to make every effort to be co-operative with other Christians; but co-operation must always be bounded by biblical strictures. As a Community of Faith, we are willing to co-operate with other Christians on the basis of doctrine and not on the basis of mission. By this attestation, I mean clearly to state that doctrine underlies all co-operation—even when we have agreed to co-operate in areas of mission. Without a doctrinal foundation, there is nothing on which we may build fellowship. Therefore, any supposed co-operation built on anything other than doctrine is a fantasy leading into ever-greater error. Moreover, each congregation is responsible to determine for itself the limits of co-operation and the degree of participation with other churches.
The practical import of this truth is that there can be no authoritative hierarchy, as such, permitted among the churches of our Lord. There are neither popes, nor cardinals, nor bishops authorised to appoint individuals for pastoral oversight. No power clique exists for regulating the pastorate through ordination or to direct the churches. Neither is there to be found in the canon of Scripture executive directors, area ministers, director of missions or other individuals holding authoritative offices to direct the affairs of the churches. In the whole of Scripture there is not to be found a single synod, diocese, convention, union nor any such entity possessing authority over the churches either collectively or individually.