The twelve disciples knew that they were special to Jesus.  He had selected them and was training them to be His apostles.  They were indeed loved and chosen (Luke 6:12-16).  But they extended that idea to feeling that they were exclusively special, and therefore nobody else had a right to minister in Jesus' Name.  Although Jesus had just rebuked them as individuals for arguing who was the greatest among them (Luke 9:46-48), they were trying to assert their uniqueness again - but this time as the group of chosen trainee apostles.  

There is a sense in which they were unique, as the future foundational apostles of the church.  They had all been with the Lord Jesus from the time of His baptism to His ascension, they had all witnessed His resurrection (Acts 1:21-22).  They were the foundations of the church along with the Old Testament prophets and Jesus Christ in whom they all trusted (Ephesians 2:20).  The apostles had the task of formulating and preaching sound doctrine (Titus 1:9).  But they had no monopoly on serving the Lord Jesus Christ.  That was to be the privilege of every believer, in different ways, in the centuries to come.

Having found a man who was casting out demons in the Name of Jesus, the disciples tried to prevent his ministry because he was not one of the Twelve.  Again, Jesus had to rebuke them.  If somebody's service is in the Name of Jesus, believing that Jesus has the power and authority to act, then they are also working in God's family business.  Only those who truly believe in the Lord Jesus as the Saviour of the world will dare to use the Name of Jesus (Acts 19:13-16).  That Name alone marks out those who are on the Lord's side.  Those who deny His Name have no place in gospel ministry (Acts 8:21).

Alas, down through the ages, the church has repeated the same mistake by 'professionalising' ministry.  Many with a good heart and sincere faith have been excluded from representing the Master of time and eternity.  Although it is important to have good order in the church, and to ensure that only sound doctrine is taught, professional ministers (people who get 'paid' to be pastors, evangelists, missionaries) should be careful to enable every member who is appropriately gifted to be trained and encouraged to have a part in ministry.  Their role should be to ‘equip God’s people for works of service’ (Ephesians 4:11-12).  The key is, 'the Name of Jesus': those who love His Name, and humbly place themselves under His authority, become disciples who need to be enabled to serve their Lord.