Or do you only need the username and password of one admin? I share a computer and we are both admins, but I am worried that the other admin could access my files via Apple Remote Desktop. Is that the case or would he need my password to access my user account and files via ARD?
It would be necessary for the admin trying to access remotely to know each password for the account. Your files would not be viewable, unless you were to specifically grant the other admin access via ARD or specific shared folder.
To remotely access an administrator account remotely, you will need the IP address as well as username and password of the specific account that you want to manage. A MacOS Mojave device would have to have Remote Management enabled, at which point the admin would only be able to administer the account to which he had been granted access.