Who can publish a space?

Only the space administrator can publish all the pages in the space in a single publishing action.

You need to have

Space administrator permission in both the draft space and the published target space
At least view, edit, and delete permissions on all the pages
No page-level restrictions on the draft space (or published space pages if previously published)

Page editors can publish on a page-by-page basis if this setting has been enabled in the space and they have the required page view and edit permissions.

You also have to be a member of the Confluence user group added to the Publishers Group. This configuration setting is managed by the Confluence global administrator in the Space Publishing - global configuration.