In the press this code, it requires users to be able to publish posts.
On line 10 (in trunk anyways) there is a line that reads:
if ( ! current_user_can('publish_posts') ) wp_die( __( 'Cheatin’ uh?' ) );
This makes it so only users that can publish posts are allowed to user press this (makes sense). But, if you want some different functionality, just change the wp-admin/press-this.php file.
You could replace 'publihs_posts' with 'level_1' etc. Consult ryan's post:
and the capabilities API: