This is rather general.
As an SM, your best friend becomes documentation. You NEED to document every discussion - the major points etc. Date and time them. Then, if you are at an impass, like you seem to be, look back through those notes.
You then have to decide if what has happened is impacting on the show.
If you look through the notes and it shows relationships are steadily deteriorating, you NEED to get someone in to rectify it if you cannot do so yourself.
Performing arts are highly dependant on trust.
You meantioned you are worried about confidentiality. You should be able to talk to the head of your department/whoever and just say "We are having issues between certain members and I need to arrange CONFIDENTIAL conflict resolution and mediation." If/When you are asked for more information, simply say "I am sorry, but I guaranteed confidence, and I will not break it."
Generally, schools and places of education are good with confidentiality. Trust between teachers and students is an important part, and a teacher breaking confidence can be disasterous.
The only times you should break confidence are when the issue could result in harm to the person in question, or another person.
I took a three month mediation course. A lot of it is a load of bull, but there are some good issues raised.
First, find a second person to 'assist' - this is essential. Then explain to them the process.
You sit down with the first person, whilst the second person sits out with your assistant. The assistant is to talk about the weather with the second person and keep their mind occupied, whilst you get the first persons point of view on things, and the issues. Then you swap. Neither of you mention anything that has occured with the other. Get both to come back in.
Sit them NEXT to each other or at right angles to each other, not across the table (that makes it seem like a battle zone). In silence, go over your notes.
Look to the party who feels wronged and ask if they would like to tell the 'wrong doer' how their actions have made them feel, then reverse the situation.
Basically, keep them talking in normal/low tones, and see if you can reach a comfortable compromise. Through this, your assistant is taking notes, preferably verbatum, or it is being recorded (both parties should be advised of the fact the conversations are being recorded). Suggest a compromise (ie "Person A would it be aceptable if person b made a public appology, we would not mention names of course.") if nothing else has arisen.
Finaly, write up exactly what has been agreed on (ie "person a feels that person b's comments on dd/mm/yyyy were inapropriate. person b has expressed his appologies and will make a public appology without mentioning names on dd/mm/yyyy to the entire cast") then have them both sign it.
Un-qualified mediation is only applicable in cases where there has not been a criminal report filed. The second it becomes a matter for either the police or the civil courts, the theatre MUST back off and seek their own legal advise.
I was involved (the SM) in a production where the director came onto an actor too strongly and it was reported to the police. However the company asked me to mediate. I talked to the local police before I contacted them about mediation, and was informed that my intervention could lead to the obstruction of justice as it could be seen as coersion on the theatres part to get the charges dropped so that the show could continue. I had the fun job of telling the uber-controlling producer this, and the shows lead and director were re-cast.