Having spent over 15 years of my life in paid, pastoral positions in a series of local churches...and now a similar number of years in voluntary leadership as an elder...I have had my fair share of experience with people who should be following leaders thinking they need to lead the leaders.
This attempt at leading the leaders often comes, not as a private conversation calmly expressing concerns, but in the form of a written missive...and the worst of those are often not signed by the author.
I was reminded of these things as I recently re-read a passage from John Stott's book, Basic Christian Leadership:
This whole passage (1 Corinthians 4:3-7) emphasizes one main point, namely, that ministers of Christ (whatever form their ministry may take) are accountable to Christ for their ministry. Of course, we must listen to human criticism, however painful it may be, especially if it is untrue, unfair or unkind. But ultimately we are responsible to Christ, and I believe him to be a more just and merciful judge than any human being, committee, council or synod.
This tells us what to do with anonymous letters. They can be very hurtful, but if the author of a letter lacks the courage to divulge his or her identity, we should not take its message too seriously. A story is told of Joseph Parker, who occupied the pulpit of the City Temple in London when C. H. Spurgeon was preaching in the Metropolitan Tabernacle. One day, when Parker was climbing the steps to his pulpit, a lady in the gallery threw a piece of paper at him. He picked it up and read it. It contained only one word: "Fool!" Parker began his sermon with these words: "I have received many anonymous letters in my life. Previously they have been a text without a signature. Today for the first time I have received a signature without a text!"
Truth is...everyone probably has something to gain from hearing their detractors, just keep in mind who you're really working for; and that's an important point no matter who signs your paycheck, because who signs your paycheck is not who you're ultimately working for.