Nothing should keep you from praying. If the request is to remain confidential, then by all means go to another room to pray. If this is unavailable at the time, then pray without mentioning the person or request. After all, God knows. Confidentiality is the key word.
All prayers are strictly confidential.
Your prayer people should not talk about the situation in conversations with others. Do not go up to the person in church concerning the prayer request unless they approach you to talk about it. Do not discuss the situation among other prayer partners. We all have problems, but do not bring them up during your time of prayer with others. Do not talk about your own personal problems when called to pray.