“the right side.” That is the south side; the biblical world was oriented looking east, so south is to the right. Ezekiel 47:1 uses different words here for “right side” and “south” in the same sentence
“house.” The house, house of God, is the Temple.
“south of the altar.” The healing water flowed out from the south side of the Temple threshold, then flowed eastward south of the altar, then flowed to the wall of the Temple just south of the east gate (Ezekiel 47:2). As it left the Temple it was just a trickle.