In Matthew 28:19-20 (NASB) is says:
Go therefore and make disciples of all the nations, baptizing them in the name of the Father and the Son and the Holy Spirit, teaching them to observe all that I commanded you; and lo, I am with you always, even to the end of the age."
Here we got discipleship, baptizing, and teaching. Converting, baptizing, and growth. Is it right for any Christian to baptize a convert if the person wants or is asked? Or should a minister or leader or a person ordained, of a church that is allowed?
Think about to whom those instructions were given.
A rag-tag bunch of peasants mostly from Galilee.
(In American terms think Kentucky Hillbilly or Mexican illegal immigrant for comparable social status.)
They had no standing or any office.
What we refer to as the Church did not even exist yet.
Pentecost, some days off, had not yet occurred.
No Scripture dealing with water baptism exactly specifies who should be baptising and who not.
The only unambiguous characteristic that shines through is that both the baptiser and the baptised should be believers committed to Jesus Christ.
True Christianity does not place arbitrary gatekeepers between man and God - the priesthood is truly of all believers.
Another aspect of baptism that does truly need to be understood is that it is a public acknowledgment of a spiritual reality. Those who do preach baptismal regeneration are truly deceived - no one is saved by taking a bath, no matter who is doing the baptising!
This is not to diminish the spiritual importance of water baptism as it the most public demonstration of an individual's faith that most will make in their lives.
I personally believe that water baptism should only be the start of publicly making declarations for the reality of the Gospel.
If one really thinks about it the first mention of water baptism in the Christian context is within the larger context of the Great Commission.
The Great Commission is simple enough - the mission of the Church is Missions!
With respect to offices in the Church: they are valid but, and it is a but, one should be careful NOT to ascribe too much to them per se. I might prefer someone with no title or office in any church to baptise people than someone with a title or an office in a church - it will depend absolutely on what I know about that individual, I personally do not care about the office or the title, or lack of it, as a marker of spiritual qualification.
Most people think of water baptism as an induction into a local church. However, as already indicated it is a public demonstration of a spiritual reality. One is already a member of the Church universal - in other words the true body of believers unrestricted by history, ethnicity, language, doctrinal difference, or membership of a local church - before the issue of local church membership becomes a relevant consideration.
(I believe in church membership, but again, church membership and attendance does not necessarily correlate well with any spiritual realities.)
The bottom line: Water baptism needs to be done by people who are committed to Jesus Christ, and the ones being baptised also need to be committed to Jesus Christ.