I am running Boinc Manager Version 7.17.5 (x64) on Windows 10 version 1909.

When I open Boinc Manager I have the following two messages:

Asteroids@home: Private message
You have received a private message .
4/17/2020 4:50:07 PM · more...

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GPUGRID: Friend request
ensotron has requested friendship with you. Please accept or decline.
4/14/2020 6:09:09 PM · more...
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When I click either one I am brought to a screen which asks for my email address and password.
I enter the name and password and it says

"No account with email address <my email address> exists. Please go back and try again."

I am logged into Boinc and I verified my email address and password by logging out and logging back in.

Do I need to create separate accounts for Asteroids@Home and GPUGRID?

If so, then how did anyone send me a message or a friend request if I don't have an account?

BOINC is designed to help scientific researchers around the world. These researchers set up 'projects' which supply work which needs to be done, and BOINC manages the flow of work between volunteers and projects.

In order for you to do work for a project, somebody or something has to set up an account for you to use. Normally, you set up your own account, using your own email address and password, and that gives you access to things like private messages and friendship requests through the project website.

From your description, you haven't done that. So I surmise that you joined BOINC through an alternative route called 'Science United'. Science United makes these project accounts for you, but does so anonymously. Although the account has an email address and a password, they are random and secret - nobody tells you what they are, and there is no way of finding out.

If you did join BOINC through Science United, I'm afraid you'll never know what was in that private message, or be able to accept the offer of friendship.

It's a strange world.