I am writing this post because a friend asked me to flush out some things in my last post. I thought it would be helpful to do that here so that, if others would like to, they can view my answers.
1) Who is a disciple? Aren't all believers disciples?
The word 'disciple' as it is used in the New Testament, means learner (Mt 10:24; Jn 18:16; Acts 9:26). In a broad sense every image bearer of God is a disciple--we are all learners of something. Peter, James, John and Andrew were fishermen and learners of their trade, the Pharisees were learners of religion, etc. We cannot help but learn, simply because we bear the image of God. Jesus mission in calling his disciples was to make them learners and followers of Him.
While walking by the Sea of Galilee, he saw two brothers, Simon (who is called Peter) and Andrew his brother, casting a net into the sea, for they were fishermen. And he said to them, “Follow me, and I will make you fishers of men.”
Matthew 4:18-19 ESV
It's the fall that ruins God's original intention for discipleship--that we be learners and followers of Him. As a consequence of the fall, we are learners of our trade, culture, the media, etc. Jesus mission in calling His disciples (and us as well) was to show them God's original intention for their life as a disciple. Because Jesus--the God Man--was the image of the invisible God, he was perfectly equipped to model what it looked like to be truly human, and thus, a true disciple and follower of His Father.
Before Jesus approached Peter and Andrew, they were disciples. During the disciples three years with Jesus, they were disciples. We shouldn't assume that the moment these men began to follow Jesus that they were regenerate believers in their Messiah. Somewhere during those three years their hearts were regenerated by the power of the Holy Spirit and the Gospel, and they became Believers as well as disciples of Christ, but from the moment they dropped their nets at the Sea of Galilee they were disciples (of Christ).
Just to reiterate--all image bearers of God are disciples of something. It is the Christian's mission and responsibility and the work of the Gospel to make disciples of Christ who build the entirety of their life around their Savior, God, and King.
2) Is discipleship (as we understand and practice it, coffee shops, one on ones, etc.) our responsibility? I ask because if we're called to make disciples, once they become disciples, does the role of discipleship then become to role of the elder?
Discipleship is the work of showing, and teaching others, through relationship, how the Gospel applies to every area of life and effectually makes them more like their Creator and Savior, Jesus Christ. It always our responsibility! If we believe the Jesus' commission in Matthew 28, then it is our duty to make disciples if we are to be obedient to our Creator and live lives worthy of the Gospel.
Now the eleven disciples went to Galilee, to the mountain to which Jesus had directed them. And when they saw him they worshiped him, but some doubted. And Jesus came and said to them, “All authority in heaven and on earth has been given to me. Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, teaching them to observe all that I have commanded you. And behold, I am with you always, to the end of the age.”
Matthew 28:16-20 ESV
Consider the previous idea that 'all image bearers of God are disciples of something'. What then is the responsibility of the believer? To make disciples of Christ. The mission of making disciples is done in the relationships we have with those who know Jesus and those who do not know Jesus. Consider the fact that the followers of Jesus may not of even known that they were being "discipled" by Christ. They were following Him, and watching Him, and learning from Him, and in doing this their hearts were increasingly attracted to Him to the point that many gave their lives for the Gospel of Jesus Christ--the one they had followed for three years. They were always disciples, even before they were believers in the Gospel.
In light of that, the better question might be: Once they become a believer isn't the responsibility of discipleship that of an elder/pastor?
My answer to that would be, yes and no. The work of the pastor is to equip the saints for the work of the ministry (of making disciples).
He who descended is the one who also ascended far above all the heavens, that he might fill all things.) And he gave the apostles, the prophets, the evangelists, the shepherds and teachers, to equip the saints for the work of ministry, for building up the body of Christ, until we all attain to the unity of the faith and of the knowledge of the Son of God, to mature manhood, to the measure of the stature of the fullness of Christ, so that we may no longer be children, tossed to and fro by the waves and carried about by every wind of doctrine, by human cunning, by craftiness in deceitful schemes. Rather, speaking the truth in love, we are to grow up in every way into him who is the head, into Christ, from whom the whole body, joined and held together by every joint with which it is equipped, when each part is working properly, makes the body grow so that it builds itself up in love.
Ephesians 4:10-16 ESV
It is the pastor's job to make disciples himself and to equip the flock of God with necessary teaching and care so that they too are able to make disciples who make disciples. We would be disobedient to the call of the Great Commission if we discipled someone, they came to faith in Christ, and then we "passed them off" to the pastor. If we believe that discipleship is done in and through transparent relationship then we would simply be being a bad friend and brother if we left the work of making a disciple once they became believers in Jesus. Jesus is a perfect friend and brother who never leaves those whom He is in relationship with (Prv 18:24). Abandoning a relationship with someone at the point of conversion so that the pastor can "take over" would be the opposite of what God has called us to. Practically speaking, if we believe that discipleship is done through relationship, than it would not be the most helpful way to love your pastor; imagine if it were every pastor's responsibility to have a close relationship with everyone in their church once they came to a saving knowledge of Jesus. Not only is that impossible, but it's a great way to ruin a marriage and torpedo a ministry. The pastor's job is to make disciples and to equip the saints to make disciples themselves.
If we believe that we are always "being saved" as Paul writes (1 Cor 1:18, 15:2; 2 Cor 2:15) than the mission of making disciples is never over for the believer. Just as the work of the Gospel in the heart of the believer is continual and constant, so are our relationships. Granted, some discipleship relationships last our entire lives while others only last for a season, sometimes we may disciple someone and never see them come to a saving knowledge of Jesus (Judas comes to mind), but the work of discipleship is never finished.
In terms of qualification and responsibility, the only thing that seems to separate an elder from any other believer is the desire to be an elder (1 Tim 3:1), and the ability to teach and shepherd the flock of God. Every other quality that should characterize an elder should also be the character of every other believer and disciple of Christ. If this is true, than it is the pastor's and "the laymen's" responsibility to make disciples of Jesus Christ.
There is certainly more I could say on this, but I simply wanted to try my best to thoroughly answer the two questions that were asked. I hope my response has been helpful, and I am more than open to rebuke, wisdom, and more questions if you have them.