You should be able to add both public and private repositories on GitLocalize. If, however, the private repository you wish to add is part of a GitHub organization, you will need to have the Owner role in that organization in order to add it.
If your repository does not show in the list of available repositories, click this link:
It will lead you to GitHub, where you need to select your organization and then grant the GitLocalize App access to the specific repository you're trying to add:
Projects added for private repositories are also private.