Ren here from the Shopify support team, hoping to help you out here.
To clarify, you're trying to publish pages such as "Contact" or "About Us" pages? What specifically are you working with? When you go to Online Store > Pages, check the boxes next to your pages and click "more actions" are you seeing something other than these options?
Just in case, I want to be sure you're not referring to actually making your website in general available to the public. In order to do this, you'll need to go to Online Store > Preferences and scroll down to the password page section. Disable the Storefront password and your website will be published and viewable to the public. Let me know how it goes!