The phrase "side of the path" is correct and usable in written English.
For example, you could use the phrase in a sentence like this: We walked along the side of the path, admiring the scenery.
Yaks grazed on either side of the path.
By the side of the path, they spied a milkweed.
Callow sat by the side of the path, laboriously pulling foxtails out of his socks.
Parents herded stragglers to the side of the path as faster cyclists careered past.
On either side of the path were verdant green crops of wheat and barley.
Patches of dark orange moss cover the rocks by the side of the path.
He points to a green standing stone by the side of the path.
On the left side of the path is a mini-meadow of perennials and grasses.
Ordinary suburban backyards and picket fences run along one side of the path; on the other — sic transit gloria mundi.
The hips on the dog roses by the side of the path have disappeared, picked off by the birds.
On either side of the path, there are lichen-coated rocks, logs sprouting mushrooms, and countless varieties of ferns.
