Combinar dos consultas en una

Si las consultas existentes hacen lo que desea/necesita, UNION hará que sea bastante simple combinarlos, algo como;

SELECT * FROM (
    SELECT is_private 0, <field1>,<field2>,<field3>, ... ,(SELECT COUNT(*) 
               FROM votes 
               WHERE message_id = m.message_id
                 AND vote_type = 'like') AS likes,
              (SELECT COUNT(*) 
               FROM votes
               WHERE message_id = m.message_id 
               AND vote_type = 'dislike') AS dislikes 
    FROM messages m
    WHERE 1 #and hidden is null
      and recipient_id = 1
    UNION ALL
    SELECT 1, <field1>, <field2>, <field3>, ... ,(SELECT COUNT(*) 
               FROM votes 
               WHERE message_id = m.message_id
                 AND vote_type = 'like') AS likes,
              (SELECT COUNT(*)
               FROM votes
               WHERE message_id = m.message_id  
                 AND vote_type = 'dislike') AS dislikes 
    FROM private_messages m 
    WHERE 1 #and hidden is null 
      and recipient_id = 1 
)
ORDER BY datetime DESC

Tenga en cuenta que debe seleccionar el mismo número/orden de columnas de ambas consultas para que la unión funcione. SELECT * hace que sea difícil verificar si ese es el caso. Si