on either hand
On either or both sides. It was quite breathtaking walking at the very bottom of the Grand Canyon, with colossal, ancient cliff faces towering over you on either hand. He strolled into the room with a beautiful woman on either hand.
on every hand
On all sides. Deep in enemy territory, we were constantly surrounded by dangers on every hand. Coming from the countryside, I find New York City quite claustrophobic, with huge masses of people on every hand.