The answer to your question differs depending on the purpose of it.
Now that the position is officially mine, is it appropriate for me to know who I was up against?
It is very important to respect others privacy. Asking for their names is definitelty inappropriate, just as trying to figure out who the other applicants were. Does this mean that you have to let this go?
If you are just curious: Yes, if you want to use the information to handle your new position with success: No.
The interviewer was well known to me beforehand and given the nature of the work I suspect the other applicants will also be familiar faces.
You can ask the interviewer what made them choose you to find out what they liked about you. Also try to ask him/her what they liked about the other applications compared to you. This way you show some self-reflection and you can work on this.
Does this change if I am going to be managing one or more of the other candidates?
I think it's rather important how you act as their manager instead of if you will become their manager. It is very demotivating if you try to be 'the manager'. Give them respect. None of you are better than the other, you just have different positions with different responsibilities and tasks. Accept them for their talents, not their 'failures' (even though it's not good to see shortcomings as failures).
It's your responsibility that everything runs smooth now. The future might tell you if you are managing one of the other applicants. Maybe they tell you that they applied for the position. If so... try to help them achieve a position like it. It's always good to help your fellow colleagues, you'll earn their respect and you give them hope.