I think you are confusing pages with modules aka feature instances. It is not possible to add a site page to MyPage, it is only possible to add modules that are marked as available for MyPage.
So a user may create a custom page on MyPage that is named the same as a site page (a page in the main menu) but a user won't see the content just because he created a page with the same name.
Now if the Content Admin create this site page "OnlyManagers" and put some feature on it like an Html module (possibly with the same name as the page). The permissions of the page have nothing to do with whether the content is available for MyPage, this is configured in module settings by a checkbox. Click the gear icon next to the title of the content instance to see the setting "Available for MyPage". If this is checked the content can be added to MyPage, if not checked it cannot. There isn't currently any way to make it available to MyPage based on roles.
Hope it helps,