Now that JIRA 7 has removed the jira-users global permission, it's not possible to add simple login permission to a Group without giving the users access to all JIRA functions.
I have searched the forums far and wide, and it seems like the solutions worked for 6.x versions.
Here's the story:
When a User that is a member of a particular Group logs in to JIRA, they can see only their project and the boards and issues related to that project.