I think it should be a preference *and* a permission. First, a permission by the owner (does the owner want this mod to be privy to all owner messages), and if so, a preference by the mod (does that mod want to receive the emails for owner messages, or just have access to them in their member Notes page).
What's still getting lost here, or seems to be, is that currently, all mods have access to owner messages by members via the member Notes page, so the permission to see such messages already automatically exists. I think it shouldn't, and the two alternatives for ways around it I see are (1) make access to member Notes pages a specific permission (something I've been asking for, and I think received some support for here, in the past), or (2) make access to owner emails a specific permission (this would entail the Notes page including owner messages only if the permission was granted).
A minor detail is what to do with owner messages from non-members in the two cases. If (2), that would be taken care of automatically. If (1) (Notes page access), it would have to dealt with explicitly.
Messages are the sole opinion of the author, especially the fishy ones.