In-game reports work only if you do not have forums, or you can simply tag them in reports, Private reports get used by people who report their friends and stay friends with them.
Sorry but Moderators are very busy. They don't really have the time to send individual messages about reports and other things.
I did when I was mod. There were quite a few people who would private message me reports, and I saw and had no problem doing the reports they gave me.
People could use the reports and ban appeal sections for reports and ban appeals.. sounds like a good idea. I do agree that mods should always keep their inboxes open however.
We should always keep the option to message us open, yes. There is a section for appeals & reports (as you most likely know). I don't think anything regarding reports/appeals should be done privately. We need to look after eachother to make sure we are making valid reports and accepting valid appeals.
Appeals should be done in the appeal section as should reports unless as you say they want to be kept confidential then yes I agree a mod should open a convo but I'm pretty sure most mods would do this anyway