How do I stop private relay messages on my MacBook Pro?

Turn off iCloud Private Relay

1. On your Mac, choose Apple menu  > System Settings, then click [your name] at the top of the sidebar.
2. If you don’t see your name, click “Sign in with your Apple ID,” enter your Apple ID (or a Reachable At email address or phone number that you added in Apple ID settings), then enter your password. If you don’t have an Apple ID, you can create one.
3. Click iCloud on the right, click Private Relay, then turn off Private Relay.
4. Do one of the following:

• Turn off iCloud Private Relay temporarily: Click Turn Off Until Tomorrow.
  • Within 24 hours, iCloud Private Relay will turn back on automatically. If you want it to resume sooner, see Turn on iCloud Private Relay, above.
  • Turn off iCloud Private Relay completely: Click Turn Off Private Relay.

1. Click Done.

For information about turning off iCloud Private Relay temporarily for a specific website, see Temporarily allow a website to see your IP address in the Safari User Guide.

If all else fails give this a try: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.