What is a delicate way of identifying key insiders?
Let me just emphasize one thing: there are very few objectively good advisors/mentors. It really depends on both the mentor and the student’s personality. Some prefer a more hands on approach while others like to be independent. Some are motivated by harsh criticism while others need a more gentle approach. 

To conclude, I think what you’d want to ask is more “what is Professor A like?” rather than whether they’re good/bad mentors.