I am working as Scrum Master and there is a team member who has made a few mistakes. He reports to another manager. I want to talk to him personally without his manager being present. Is that okay administratively and (more importantly) ethically?
Sure it's ok - and it's your job to keep the team working and remove impediments. This includes the team member who made a few mistakes.
Due to the fact that you asked the question, I assume you don't have a perfectly open and failure tolerant culture in your company or team (who has?). Because of that I will try to give you some hints on how to prepare the talk.
Before you schedule the talk, think about what you appreciate on your collegue and what he/she adds to the team.
This should enable you to not be focused on the mistakes but to be supportive and open minded.
When you schedule the talk, think about how to put your collegue in a comfortable surrounding / mood, e.g. by extending a common coffe break.
Nevertheless, during the talk speak honestly about your observations. Make clear statements about what you assume and what you observed. Within negotiation theory there is the expression: Be hard on the deal, soft on the people. This counts here in quite a similar way also.
The goal should be similar to a retrospective / lessons learned session:
- What is the root cause of the mistake?
- How to avoid the (specific) mistake in future?
- How to avoid similar mistakes by others?
Of course, it should be ok. If you have doubts about conversation, it means that something wrong with your working process.
I am concerned about two things in your question:
Why do you have additional manager in your process? Scrum prescribe only two managers role: Scrum-Master and Project Owner. Scrum Team should have only one Scrum-Master and Product Owner. Your responsibility as Scrum-Master to prevent intervention of another persons to working process. You should have enough authority (formal or non-formal) in your company to do this.
Why developer should report to you? Scrum-Master not a boss of a team. He can't command developers. He is coach and helper of team (his responsibility are facilitating Scrum events, removing impediments and so on). But team should be self-organized and take all decision by themself. If team asked you to gather reports for more transparency (or you advised to team to do it, and team agree with it), and then one of team member didn't report you, you should not have any doubts to ask this person why he did not complied with the decision the team.
Tobias has a great answer for standard agile.
As DB pointed out though, it sounds like you're possibly running in a non-standard model.
If you are a people manager (you write job reviews, hire/fire) then that supersedes your role as a scrum master. You need to talk to your HR to find out the policies at that point for a manager talking to another persons reports.
For the record, people managers being scrum masters of people that report to them is generally not advised. It creates all sorts of issues, like the one above.