Priests in my archdiocese do two full days of training every single year. Each parish and school has a safe environment director other than the priest to monitor it and see that all employees and volunteers are certified (you won't find this listed as a job as it is usually assigned to someone with another title. In my parish, it is typically the director of religious education; at schools it is a teacher or lower level administrator--those people also have extra training). All dioceses are also required to do an audit annually which is done by third parties. Safe Environment programs are also developed by third parties. Our diocese uses one developed at Boys Town (which is a real place...ran into someone once who thought it was just an imaginary movie place!).
And, yes, employment files are generally private. Priests are employees. They get salaries and insurance just like other employees. There was definitely an issue; I'm not denying that at all. But since 2003, the church in the U.S. has taken a lot of concrete steps to fix it. People seem to enjoy assuming they have not. The rules are followed strictly, too. To the point that it is difficult sometimes. As a teacher in a Catholic school., there were tons of things that I knew my peers in public school did regularly that we could not even think about. It is far more stringent than the public schools I have worked in. One large district near me doesn't background check any employees. They have had multiple incidents of support staff (janitors, bus drivers, etc...) fall through the cracks in the system (which are frankly canyons, not cracks) and be hired with criminal records involving children then abuse kids in schools or extracurricular activities. Meanwhile, at my parish, a volunteer was removed in the middle of the religious ed year a couple of years ago due to an arrest. Because everyone is supposed to be monitored by the designated person in each parish or school--so if they are doing their job, anyone who has an arrest or charge won't continue even if between required checks. 13 years ago, a friend asked me to help her with a summer program at her Presbyterian church. I asked if I would be able to as it was rather last minute and there wouldn't be time for me to undergo a background check or any required training for safe environment. She had no idea what I was talking about as it wasn't something they did.