Google has released an update that allows you to choose exactly who can view your content in the new Google Sites. To publish your site for a specific audience, select Share with others and change Published settings to Specific people can view when published. Next, add the individuals you want in the Invite people box and select Can view published.
The new feature is now on the web. Google Drive Android and iOS apps will have it rolled out in the coming weeks. For more info on publishing in the new Google Sites, visit the Help Center.