This is a very good question and an important one in a time when many stories come out of sexual abuse of especially young people and women in church by church leaders. Under the hashtag #churchtoo, there are unfortunately many painful examples.
Listen to your intuition
That is why I want to start by saying something important. If you wonder about this because you are a girl and you felt uncomfortable when a male youth leader prayed with you in private, I want you to take that feeling VERY seriously. We cannot always know with our reason why something is wrong, but often we can feel it. God has given you intuition and often speaks through it. You should listen.
Depending on the relationships you have with people in church, do what you can to protect yourself. Avoid being alone with this youth leader. If he asks to pray with you in private, decline politely. If he doesn’t ask, or insists, you may have to say ‘no’ very clearly or walk away. If possible, where there are other trusted adults in your church, talk with them and explain them your concern. Remember that nobody should ever force you to be alone with a man and do something spiritually intimate like praying if you do not feel comfortable about that. Imposing on somebody, even in prayer is never the way of Christ. And as a young sister, you deserve all the respect and protection of your Christian brothers.
Rules in the church
But that does not answer the question of whether that is allowed. Many churches have rules against this. If that is the case in your church, the answer is ‘no’. The youth leader should obey the wisdom of the church leaders in this. If you really want to pray with somebody, he can help you find a woman to pray with you or invite another person to be present while he prays with you.
Guidance of the Bible
However, if the church has no such rules, we have to consider the guidance of the Bible for ourselves. Some important guidelines are that we should avoid even the appearance of evil (1 Thessalonians 5:22) and that church leaders should be above reproach (1 Timothy 3:2), and sisters in Christ should be treated with absolute purity (1 Timothy 5:2).
Wrong speculation or accusations
Even if a youth leader means well, he can invite all kinds of wrong speculation or accusations if he spends private time with a girl. Also, he can put himself or the girl in the situation where sexual thoughts or desires interfere with a healthy relationship of spiritual mentorship. For that reason, I think a male youth leader should always try to find a woman who can pray with the girl or only pray with her if there are other people present, preferably another adult woman. If in a particular situation this is not possible, and a male youth leader is asked by the girl herself for prayer, he should do so in a public place, where others can see him, and avoid all physical contact. However, I feel he should never initiate personal prayer with a girl himself.
Also, youth leaders of both sexes should be careful to make sure that their prayer is something the girl or boy really wants and needs, and not just something the youth leader wants. Young people may find it difficult to say ‘no’ to an adult, so the adult should be extra sensitive. Again, imposing is not the way of Christ.