SQL Server – Login Failed for User with Read Access

One of our user from another domain (...)

To authenticate against a resource in another domain, either there must be a domain trust established (ask your Windows sysadmins), or pass-through authentication must be used (very likely not in this case). This is true for Windows in general, not just SQL Server.

Do I need to setup my SQL Server 2008 Enterprise in order to allow domain user (...)

If there isn't an existing Windows user or group login for the user, then yes, you'll need to create a new login.

Kindly note that there might be more than 50 domain users that may required to access the web application and therefore it would be inefficient to add each domain user to the database's user rights

I agree -- with this type of scenario, I would recommend setting up a group in AD, and then creating a Windows group login in SQL Server. This way, access is controlled by AD group membership, and you don't have to mess around with changing logins in SQL Server once it's set up.

Add BOTH Domain\Servername$ login and NT AUTHORITY\Network Service login to SQL Server instance giving you above problem and give both logins the right permissions.

If you are working in a Spanish environment or another language try adding NT AUTHORITY\Servicio de Red or whatever the equivalent is in your language.

For steps on adding logins see links: http://support.microsoft.com/kb/889646/en-us and http://www.c-sharpcorner.com/UploadFile/specialhost/how-to-fix-login-failed-for-user-nt-authority-iusrs-in-share/ and https://stackoverflow.com/questions/2806438/login-failed-for-user-domain-machinename

If you have a SQL Server Instance with logins that have been working for you when accessed by application pools and the like, you might want to check which logins are not in the troublesome SQL Server instance and that you might need.

Other comments:

Ok, even though the error said: Login Failed for user 'Domain\Servername$' and I added that login to the problematic SQL Server instance per what I'd seen described on the internet, it seems that the login I ALSO needed to add was NT AUTHORITY\Network Service.

I basically decided to add any logins that were in my normal SQL Server instance that I was using for the regular SharePoint databases and that weren't in the SQL Server instance I was using for SharePoint Reporting Services to the latter. Something else that tripped me up when I was looking originally to add the NT AUTHORITY\Network Service login was that I couldn't find it. It turns out I had to add it in Spanish as NT AUTHORITY\Servicio de red, which is wierd because the app pool has its Identity in English. I guess this happens when some systems have English as default language and some others have Spanish