In my occupation (which I've managed to never talk about here, til now) I am the buffer between our Delivery team and the client's Operations and Delivery teams. I have no direct reports though all communications go through me for my environments.
Same kind of deal here, Henry...I filter out crabby dev/tech emails, direct pissed customer calls and emails to the right tech lead or mgr and generally try to keep things moving. The one advantage I have is that state of the environment is full disclosure, so when I've seen/heard too much on one subject flying through I get to call what I refer to as an "All the liars in one room" conference. At that point I get to formally assign things out to leaders and I generally never hear any more about it. It seems like you have to do much more of a balancing act than I do. Hats off, my friend.