Content (images, text, links, buttons etc) that you add to the 'site sidebar' appears on all pages of your website (unless you have manually disabled the sidebar on a page via page properties - page layout).
The 'page sidebar' section displays content only in the sidebar area of the page it is added to.
To move elements from one section to another, simply click the up/down arrow at the top of each element to place it where you wish.