Screens connects to another computer using an existing user account on that computer, so when adding a new screen to your Screens Library, enter the username and password for an account on the destination computer.
Your Screens ID doesn’t provide authentication services, and your Screens ID credentials are never sent to a computer you connect to. Your Screens ID only serves as a repository for information about the computers you have Screens Connect installed on. Simply put, it works like a phone directory, providing Screens with an up-to-date IP address and port number at any given time for the destination computer you want to connect to. Using your Screens ID credentials in a screen’s settings will result in an authentication error such as NSLocalizedFailureReason=User “firstname.lastname@example.org” doesn't have access or the username or password is incorrect.
This may be easier to understand if we look back to the dumb terminals of past decades. A dumb terminal didn't include a lot of intelligence — just enough to connect to a server to download data. It essentially acts as a doorway to another computer. Of course, the server can’t let just anyone in, so a username and password are sent from the dumb terminal to the server to gain access.
Screens is very similar. It simply opens a connection to another computer to download what it needs, namely a graphical copy of the other computer's desktop. Screens will always connect to another computer using an existing user account (or macOS' Guest account) on that computer.