I am using the domain access module in Drupal 7.
I want one domain to be accessible by anonymous users and the other domain to be accessible only by authenticated users.
How can I give the "view published content" permission to anonymous users for only one domain.
Is seems that Domain Strict, a Domain Access submodule, may help you, as stated in this issue:
DA is a node access module (which is designed to enhance what 'access content' means), and you should be able to do this now by using Domain Strict and configuring user behaviors.
Domain Strict forces people to be members of a domain in order to view its content. You would enable the module, then you will note:
1) Anonymous users cannot access any content. This is because they cannot be assigned to a domain (since thiey have no account).
However, you can use the User Defaults setting page to assign anonymous users to the primary domain.
2) Use the same User Defaults setting to assign all 'authenticated' users to the domain(s) you prefer.
Under this setup, anonymous users would still have the 'access content' permission, but can only see nodes posted to the primary domain.
See section 4.10 of README.txt and the README for Domain Strict.
Check also README.txt from Domain Strict.
[Domain Strict] it changes the default Domain Access behavior. This module makes the grants given to a user specific to the domains that the user is registered to see.
You'll have to deal with User Defaults setting to set permissions for anonynmous users since initially they don't have domain assigned.
