I work for a church. I have gotten all sorts of anonymous complaints. Phone calls and handwritten notes are most common. Any time you change anything you get complaints from somewhere. It is difficult to lend them and credibility if they are anonymous.
But what if the complaints are public, and anonymous? How do you handle that? it’s difficult to do biblical conflict resolution when you can’t go to the individual. I think the best way is to address concerns that are legitimate to the congregation, and ask the person to come speak directly with you, or a rep from the church. If a news story emerges, then answer honestly, and say what you can. When you are walking in the light, it is simple speak the truth. If you have made a mistake, own up to it. If it’s just a complaint, address it. Diffuse the story with the truth.