With so many teachers making the news for sexual abuse of students on a regular basis, how is it possible to trust the teachers when you send your child to school? With so many abusive day-care workers, sports coaches, doctors, police officers, babysitters and neighbors, how do you trust anyone?
The answer in each case, including that of priests (and ministers, rabbis and clergy of other faiths), is that the great majority of people in the profession are not abusers, and you keep your eyes open and take appropriate precautions, including reporting, against those who raise any red flags. Also, organizations and professions that work with children and teens generally now have some level of heightened standards and procedures for abuse prevention.
As for allowing my child to be alone in a room with a priest, priests nowadays are generally forbidden under safety policies from being alone with minors, therefore I would not expect any such alone time to happen and if it did it would be a red flag.