It can be challenging or even scary to try having difficult discussions with friends or family, let alone with colleagues or students. However, these discussions are necessary to learn about and understand how people feel. This is often the first step in trying to bring justice, or change, to our departments.
You might also want to know more about leading difficult discussions if you are thinking of leading diversity trainings at your institution. This website helped me think through and prepare in advance when I was about to do such a training.
Something I have found useful also for trainings is to try to think in advance of what things people might say that might be difficult to respond to, hurtful, uncomfortable, etc. And try to think of things you can say, if only to change the subject if you do not know how to respond. But really there's nothing like practice, and also there is no such thing as fool-proof preparation, so try it out! Also, taking trainings yourself (maybe some on-campus group offers such things, even if not tailored to math) can help build your confidence.
And last but not least, conflict resolution or mediation training could be helpful (I have found it to be a useful life-skill in any case, and it has made me a better listener), often community groups or even on-campus groups offer these. An example is this course offered at Harvard. You may also be able to find a community- or school-based association that offers mediation help and/or training near you.