Paul's letter to Titus sounds very much like his first letter to Timothy. In fact, both Titus and Timothy were planting churches and facing the same challenges.
In the letter to Titus, however, Paul spells out who ought to be teaching whom. You will notice that men teach men and women teach women. There is a good reason for that.
A teacher in those days spent considerable time with the pupil. Jesus taught his disciples by living with them, day in and day out, for three full years. He invested Himself in them. He gave them all He had to give, and they loved Him in return.
When a teacher invests himself in a pupil, a close relationship grows between them. Can you imagine the havoc that close teaching relationship might wreak if teaching were done between the sexes? It would be an open temptation to many and could cause much damage to the church. That's why Paul separates the teaching by gender.
In my work, I talk to many people by long-distance phone. One man has called several times and explained his problems to me. He is a man who wants to be good, but he is living out of accordance to the will of God. After our last phone conversation, I realized that he was looking at me as a teacher. So I wrote him a short note to encourage him. But in that note, I suggested he find a local man to mentor him. He needs an older and wiser man to rein him back into submission to the word of God.
You cannot claim the word of God and live against its teachings. That is self-deception at its worst. If you believe in God and want to claim His promises, your actions must conform to what He has taught.
Look at Paul's list of virtues to be taught.
Older men are to be temperate, worthy of respect, self-controlled, sound in faith, love and endurance.
Younger men are to be encouraged in self-control, integrity, seriousness and soundness of speech.
Older women are to be reverent, not gossips, nor drunkards, but teachers of what is good.
Younger women are to be taught to be loving (to husbands and children), self-controlled, pure, industrious, kind, and submissive.
We can successfully be taught all these things because the grace of God assists us. We can teach these things because the grace of God gives us that authority. Paul says "encourage and rebuke with all authority." When you have learned what God would have you teach, you can speak with boldness and authority because God is the author of what you are teaching.
Oh, and by the way, we are all called to teach.
Help us not to stumble when we teach what we have learned of You. Keep us on the right path so that our words are supported by our deeds. Remind us that we are what we are only by the grace You so freely gave.
In Jesus' most precious name.