Have you ever been in a situation where a staff member, or staff members, continually raised their personal issues and complaints in team meetings or in front of other staff members? Since this situation is not that uncommon it is fairly certain that you have.
As a facilitator and former mediator, I witnessed this a great deal. The truth is, sometimes the intention is very innocent and not to embarrass, challenge or create a scene. In these cases, the person will often respond favorably to a gentle re-directing comment. "That sounds like a very important question you have, and one we might be able to better address one to one. How about you and I connect immediately following the team meeting?" will usually be met with a positive response and help to bring the meeting back to its initial focus.
However, there are other time where a person does have a bone to pick with a team member or supervisor and raises a personal issue in front of the team. The same type of statement could apply here, "That sounds like an important issue that perhaps you and Sue could talk about following the meeting." Sometimes there is value in checking to see if the issue is just one person's concern. Sometimes the group has a concern but only one person feels comfortable or has the courage to raise the issue. It is helpful to manage your emotions and response and to think carefully about how to proceed. If you notice a pattern, for example one team member constantly raises his/her concerns at team meetings instead of one to one, that would be a perfect topic for a one to one feedback discussion after the meeting. It would be very helpful to understand why the individual feels the need to voice and address issues in that way. There may be an opportunity for the "coaching conversation", a dialogue around alternative ways to address concerns.