¿Qué tal
SELECT * FROM
(
SELECT sodnik_1 as value FROM prihodnji_krog WHERE file_id='8778' AND sodnik_1 != ''
UNION
SELECT sodnik_2 as value FROM prihodnji_krog WHERE file_id='8778' AND sodnik_2 != ''
UNION
SELECT sodnik_3 as value FROM prihodnji_krog WHERE file_id='8778' AND sodnik_3 != ''
UNION
SELECT sodnik_4 as value FROM prihodnji_krog WHERE file_id='8778' AND sodnik_4 != ''
) x INNER JOIN
(
SELECT value FROM notification_sodniki WHERE user_id='16'
) y
ON x.value = y.value
Cuando usa subconsultas en un FROM
cláusula, es como si estuviera creando tablas temporales. Luego, con el alias, puede referirse a ellas y hacer una INNER JOIN