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¿Cómo obtener el recuento de la unión INTERNA y el recuento de todos los elementos?

Si hace que sus dos consultas sean compatibles con Union, puede combinar los resultados y agregarlos con una consulta adicional:

    SELECT payer, 
           sum(churned_accounts) AS "churned_count",
           sum(total_accounts) AS "total_count" 
    FROM (
      SELECT CAST(payor_id AS CHAR(50)) AS "payer", 
             count(*) as "churned accounts", 
             0 AS "total accounts" 
      FROM paid_users_no_more 
      INNER JOIN paid_users 
        ON paid_users_no_more.user_id=paid_users.user_id 
      WHERE paid_users.payment_start_date NOT BETWEEN '2015-08-01 00:00:00'::timestamp AND '2015-08-30 23:59:59'::timestamp 
      AND paid_users_no_more.payment_stop_date BETWEEN '2015-08-01 00:00:00'::timestamp AND '2015-08-30 23:59:59'::timestamp 
      GROUP BY paid_users.payor_id

      UNION

      SELECT CAST(paid_users.payor_email AS CHAR(50)) AS "payer", 
             0 AS "churned accounts", 
             count(*) AS "total accounts" 
      FROM paid_users 
      WHERE paid_users.payment_start_date NOT BETWEEN '2015-08-01 00:00:00'::timestamp AND '2015-08-30 23:59:59'::timestamp 
      GROUP BY paid_users.payor_email
    ) as All_Accounts

Tener el 0 AS "total_accounts" y 0 AS "churned_accounts" significa que las dos consultas tienen los mismos campos presentes, lo que hace que UNION posible.