I am trying to allow logged in users to read private post through the api issuing the simple api call /wp-json/posts/1 (which is a private post).
However even when I am logged in with admin rights I still get a 401 unauthorised error.
Is what I am trying to do possible with this authentication?
You’ll need to either send the X-WP-Nonce header, or the _wp_json_nonce query parameter with your request. You’ll need to generate this on the server with wp_create_nonce( 'wp_json' ), which you can then output on to the page for your script using wp_localize_script.
(This assumes you’re accessing the data from the site’s frontend, not from an external site.)