I've been thinking about adding this feature for years, but I kept putting it off as too hard and too low priority. I've finally gone ahead and added it primarily to assist the admin team with discussing submitted change requests etc. However, I'm sure it will find some good use outside of that.
You will now see a little mail notification icon in the topbar, showing a number of unread private messages if you have any.
Click it (or use the menu link under ME > Private Messages) to see all your messages. Your inbox will be displayed with unread messages highlighted.
There is a limit to the number of private messages you can have in your inbox. It's currently 100 for Free Plan members and 500/1000 for Pro/Designer Plan members. There's a good chance these might be increased later, but I wanted to start conservatively. You may get some automated messages from the Rebrickable Bot (e.g. when you get a new badge), and these are not counted towards your total quota.
Messages that you send can be viewed in the Sent Messages page, and these are automatically deleted after 6 months.
For now, given the potential abuse such a system can have, I have restricted the sending of new private messages. Only users with a minimum Level 5 are able to send new messages. However, anyone can reply to a message sent to them. I will definitely lower this eventually, after I have had time to assess its use vs abuse.
There is also throttling and spam protection in place - i.e. you cannot send too many messages per day. Like commenting, these settings become less restrictive as you increase your Level.
You will find two new settings in your profile:
So you can completely turn it off if you prefer. If you have disabled the receipt of private messages, anyone trying to send you a message will be told you are not receiving them. Note that some automated messages might still appear in your inbox but no emails are sent from those types of messages unless you have the specific notification option enabled for it (e.g. new badges).