I'm glad this is being discussed because this just came up for me this morning. A rather sensitive message I sent to a subscriber (as me, not as the owner account) is now in the activity log for all moderators and owners to see, and I strongly prefer it not be there. Similarly, I don't need to see any other private messages sent by other owners or moderators.
If I had chosen to send this as the owner, I can see why one might want it to be logged, however.
