In the new version of Google Sites you can now specify individuals who can view your published Google Site. This option is a middle ground between making your site public and keeping it private. You invite people to view your published site by entering their email addresses in the "invite people" box (found in the upper-right corner of your site editor) and then choosing "can view published" after entering email addresses.
Like almost all new features added to G Suite products, this new Google Sites privacy option will be gradually rolled-out. Depending upon your domain, you might see it today or you might not see it until the end of February.
Google Sites is featured in lesson 9 of my online course G Suite for Teachers.