I just gave this a quick test here on meta and see that my TL0 test user account does not have any UI to access messages, even after I have sent it a message and I have used it to send a message to moderators.
Meanwhile, that user can go to https://meta.discourse.org/my/messages to access their inbox.
Now that the messages UI has changed and there is simply a “My messages” link in the sidebar, I think it would be worthwhile making a change here to provide that same link when a user has messages. There can also be a messages filter in the notifications menu as there is for people who are allowed to send messages, as another way to get to your messages. Here’s how it looks now for my test user. Eventually that notification will disappear off the bottom and I will lose my easy access to my messages.
