I agree. And when I agree, I don't trust myself, so I searched it in
The
Corpus of Contemporary American English (about 450 million words):
each side of the: 524 instances (implying "both" all those I glanced)
either side of the: 1281 instances (implying "both" most of those I glanced)
The
British National Corpus (about 150 million words)
each side of: 316 instances ("both", most of them)
either side of the: 900 instances ("both", most of them)
But, what do I know? Maybe their meanings elude me.
I said it
a diestra y siniestra ("left and right; all over the place" in Spanish), a little less now Dexter ended.