Publish Permission Settings
A Google Site can have more than one editor. By default, Site editors can publish, change permissions and add new people.
The Site owner can restrict these options so that only the Site owner can publish, change permissions and add new people. But this option is a bit hidden away.
First you need to click on the 'Share with others' icon
Then you will see the sharing dialogue. In the top right corner is a settings cog
Clicking this cog reveals the setting controlling who can publish etc