Much depends on the therapist and their training and experience with clients who make suicidal gestures or self harm. Some will take it in stride and actually not give it much attention. Others will take it quite seriously and insist on hospitalization which could be an over-reaction if it was just a gesture.

Your questions doesn't seem to be asking about how therapists deal with clients who are truly suicidal and/or seriously harm themselves which is what I based the above answer on.

I was never one to make suicidal gestures and I didn't self-harm. If I made a suicide plan or an attempt, it was not a gesture at all and my therapist and pdoc were very aware of that, so they took any discussion of suicide seriously, but we had really good lines of communication particularly after some time working together. We all learned to distinguish between simply having really down, dark thoughts vs. suicidal thinking vs. suicidal ideation vs suicidal plan. How they responded depended on where I was in that range of thinking. They weren't too concerned if I was just really depressed or even when I had suicidal thoughts, but when I they saw it turn into ideation and rumination their level of concern increased because I could act very quickly and impulsively at that point, and their response changed proportionately. I knew when I was in that range and in trouble though and was pretty willing at that point to do what I needed to do to stay safe, including hospitalization.