“water was flowing out from under the threshold of the house toward the east.” Ezekiel describes the river that flows from the Temple to the east and down to the Dead Sea. For some reason it does not mention the river that flows from the Temple toward the west and into the Mediterranean Sea; that river is mentioned in Zechariah 14:8. This river is also mentioned in Joel 3:18. Joel says the river flows through the Valley of Acacia Trees. Although we cannot identify that exact valley by that name, from its name as the Valley of Acacias we know it is a valley that leads down to the Arabah, the dry area just west of the Dead Sea.
“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.