Possibly one of the most important things to get right with any group discussion, but especially a Socratic Seminar is to create a culture of respect. Students need to understand what is expected of them before, during, and after the seminar and be equipped to be able to disagree with others' thoughts while not disrespecting the other person or making them feel that their contribution was not worthwhile. This can be done by creating a set of “safe discussion” criteria that becomes part of the feedback on the session and can be used by an outer circle to help protect each other.
By providing sentence starters we can also help our students to acknowledge and confirm others' contributions before disagreeing or probing further to gain a better understanding of their position. Training and demonstrating discussion control are also important. Making sure that the person running the discussion, whether it is you or a student, is strong enough to stand up and say something when the discussion becomes personal, or when contributions are being discarded without consideration will also help to ensure that the discussion works well.
Personally, I find reminding the students regularly before, during, and after the seminar, that the goal is not to be right or prove someone wrong, but instead to deepen understanding helps to minimise these issues and keep the conversation focused on everyone deepening their learning.