This is quite an easy one. You get the following variables available in page.html.twig:
General utility variables:
- base_path: The base URL path of the Drupal installation. Will usually be "/" unless you have installed Drupal in a sub-directory.
- is_front: A flag indicating if the current page is the front page.
- logged_in: A flag indicating if the user is registered and signed in.
- is_admin: A flag indicating if the user has permission to access administration pages.
Site identity:
- front_page: The URL of the front page. Use this instead of base_path when linking to the front page. This includes the language domain or prefix.
Page content (in order of occurrence in the default page.html.twig):
- messages: Status and error messages. Should be displayed prominently.
- node: Fully loaded node, if there is an automatically-loaded node associated with the page and the node ID is the second argument in the page's path (e.g. node/12345 and node/12345/revisions, but not comment/reply/12345).
Solution
{% if logged_in %}
// Do your thing
{% endif %}