This is a very unprofessional thing to do! When we are in a Intimate situation like having your hair done ,things can come out ,I know I have unloaded my thoughts to my hairdresser and also know some of their thoughts and problems,its the nature of the job,but you would expect confidentially and also not to comment on others,it was totally wrong on all levels,and you were right to be upset about this!
You can of course leave and go to another hairdresser,but if that would be difficult for you ,this is what I would do,if this lady is not the manager,slip in and ask for a quiet word with the salon manager and ask for a private word with him or her,explain what happened and that it made you feel uncomfortable,if it is the manager,then I would write a note and address of the the manager,saying the same thing.
If the manager is unaware,they need to know as it could affect business and the person being spoken about could hear it or be told it,then you could have quite a scene.
They are quite strong things she is saying,drink,drugs sounds like she has a personal beef with this person!
Your not being sensitive ,your being kind,it's not nice and not called for !