Authenticate current user to REST API

I solved my problem.

This got me on the right track:
https://developer.wordpress.org/rest-api/using-the-rest-api/authentication/

You can just make an api call via frontend and jquery.

$.ajax( {
    url: '<?php echo esc_url_raw( rest_url()) ?>' + 'your/endpoint',
    method: 'GET',
    beforeSend: function ( xhr ) {
        xhr.setRequestHeader( 'X-WP-Nonce', '<?php echo wp_create_nonce( 'wp_rest')?>' );
    }
} ).done( function ( response ) {
    console.log( response );
} );

If you want to outsource the script in an separate file you have to use wp_localize_script
to pass the url base address and the nonce.