The team I work with is from Visitation School, an independent catholic school, so I think we’re the sort of team you’re looking for info from!
I don’t know about teachers and other school employees, but for us mentors, personal beliefs have never even come up. As far as I know, no one at the school as ever asked a mentor their religion - I certainly haven’t been asked, and I’ve been doing this for the team for 13 years! And, frankly, religion never comes up within the team, it’s never been an issue.
The school administration does have some hoops for us to jump through, though, which essentially duplicates YPP. From our mentor handbook:
New mentors will be provided with Archdiocesan required, E3 Volunteer paperwork to fill out. This includes (1) a form authorizing a background check, (2) scheduling & completing Virtus training, and (3) a form acknowledging the code of conduct.
All mentors must keep their Archdiocesan Virtus training and modules up to date, and may periodically be asked to complete another background check.
They perform background checks every few years, and updating your Virtus training (essentially a 2-hour long in person YPP program) is about the same. The code of conduct is just like what it sounds - be a good role model and don’t do anything stupid. None of it is overly religious in nature.
As mentors, we also enforce school standards with the students, which really isn’t a big deal. It means some of it is written into our handbook with the team dress code and student expectations like “You are expected to exude a respectful, Salesian, graciously professional attitude and practice coopertition, honesty, and integrity at all times.” (We make an effort to marry FIRST’s core values with the schools, it works out pretty well), and we then hold the students to it.
All said, it’s maybe a little more involved than most public school teams, but nothing that’s harmed our ability to recruit mentors - we currently have 10 mentors working with the team, and have had another 20 that went through the process, mentored for a while, and for one reason or another stopped (many due to significant life changes, like getting married, having kids, or moving to a different state).
Please feel free to PM me if you want to share/discuss more details in private, otherwise I’m happy to continue to respond in public