A possibility is, that you add via plugin a filter, there check the rights and only users with his User ID in each post can change this posts.
The follow example doing this, but you must enhance a check for your custom post type, if you will only allow this on your CPT.
/**
* Plugin Name: Display only own posts
*/
add_filter( 'pre_get_posts', 'fb_pre_get_posts' );
function fb_pre_get_posts( $queryobj ) {
if ( ! current_user_can( 'edit_users' ) )
$queryobj->query_vars[ 'author' ] = get_current_user_id();
return $queryobj;
}