Posts to have category specific sidebars with the widgets

Create a child theme (if you haven’t already) and register two sidebars:

function wpse_151695_register_sidebars() {
    register_sidebar(
        array(
            'name' => 'News',
            'id'   => 'sidebar-news',
        )
    );

    register_sidebar(
        array(
            'name' => 'Blog',
            'id'   => 'sidebar-blog',
        )
    );
}

add_action( 'widgets_init', 'wpse_151695_register_sidebars' );

Then in your sidebar.php:

if ( is_singular() ) {
    // For single posts, show the sidebar relevant to the category it's assigned to.
    if ( has_category( 'news' ) )
        dynamic_sidebar( 'news' );
    elseif ( has_category( 'blog' ) )
        dynamic_sidebar( 'blog' );
} elseif ( is_category() ) {
    if ( is_category( 'news' ) )
        dynamic_sidebar( 'news' );
    elseif ( is_category( 'blog' ) )
        dynamic_sidebar( 'blog' );  
}