Your team members must be accessing the repository using SSH & for that they have to have their ssh key mapped with github account. This will work if they map their ssh key with github account and also the repository has public rights, which they want to access.
Two steps:
1. Login and click "Invite someone" in the right column under "People". Enter and select persons github id.
2. It will then give you the option to "Invite Username to some teams" at which point you simply check off which teams you want to add them to then click "Send Invitation"
Alternatively:
1. Get the persons github id (not their email)
2. Navigate to the repository you would like to add the user to
3. Click "Settings" in the right column (not the gearbox settings along the top)
4. Click Collaborators long the left column
5. Select the repository name
6. Where it reads "Invite or add users to team" add the persons github id
7. An invitation will then be e-mailed.
It is a simple 3 Step Process :
1) Go to your private repo and click on settings
2) To the left of the screen click on Manage access
3) Then Click on Invite Collaborator