If you are unable to give them what they want or must decide something adverse to their interests, face to face conversations at least reassure them that they have been heard and are respected. They may not agree with you but are more likely to continue to respect you if you explain and are accountable for your decision. It allows both parties to exchange ideas and to clarify messages that might lead to misunderstanding. The alternative of relying on others to carry your messages runs a substantial risk of miscommunication and misunderstanding. You will be amazed by what you are alleged to have said once the message has gone through two or three people.
Personal communication sends a message of respect as well.
Just as your own words and actions will be misreported, so will the words and actions of others. Do not spend a lot of time or emotion reacting to reports based on hearsay. You may hear that Professor X said something inappropriate or is upset about something or did something unbelievable and later find out it is not true or the report has been altered substantially in the retelling. Wasting emotional capital or time on such reports until you have had a chance to confirm them is something you cannot afford to do. Often you will be cast in the role of mediator between competing groups or individuals.