"to either side" or "on either side" commonly means "on both sides"
"Either" has two meanings:
- verb-wise, it separates different options (you can have either X or Y)
- noun-wise, it refers to two similar groups (there was no light on either side of the bridge, or, conversely, the bridge was lit on either side)
(Native speaker) i read either in the sense of logical or, so one side alone (tegardless of which side) or both sides at once.
Interesting how varied the ohrasing can be read, though!
Indeed. "On either side the river lie / Long fields of barley and of rye" —Tennyson