ModMail is a Discord bot which main purpose is to provide user-to-moderator private communication. This bot needs to be hosted, as it is meant for a single Discord server.
Documentation: https://modmail.phodit.xyz/
When a user DMs the bot, a channel that only the staff can see is created in the server. From there, mods can discuss and reply to the conversation all within the same channel, getting some fancy logs when the thread is closed.
Please, read the documentaion at https://modmail.phodit.xyz/ and check the installation section.
You can check the commands here.