Either I’m missing something, or we’ve become very confused about the proper use of the word either.
We still get it right when used in the way I just used it, either/or; it’s either one thing or another; however, either is often being incorrectly used as a substitute for both. Not only is this wrong, but it can be terribly confusing.
One of our readers mentioned in the Pet Peeves page that it’s often used incorrectly on HGTV, and she is correct. For example, there are many instances when a designer puts two night stands in a room, one to the left of the bed and one to the right, and the narrator says a night stand was placed on either side of the bed. Since they’re showing pictures of the incredible bedroom makeover–so good, in fact, that the homeowner is sobbing with joy–it’s easy to understand what the narrator meant, which was a night stand was placed on both sides of the bed. If a table were placed on either side, as was stated, then there would be one night stand in the room, and it wouldn’t matter which side of the bed it was on–it would be either to the left or the right of the bed.
When speaking without the luxury of television cameras, it’s important that we use either when we mean either and both when we mean both; otherwise, there will be legitimate confusion about the number of items being discussed.
–Paul
Paul’s book–Literally, the Best Language Book Ever
Everything Language and Grammar 
Such usages are perhaps a bit old, but they’ve been standard for a long time. The OED’s first definition of “either” is “Each of the two”, and they have attestations from the late 800s to 1842. Searching through Google Books for “on either side” yields 766 hits before 1800, 16000 hits between 1800 and 1900, and almost 10000 hits since 1900. And while I’m sure some of those are “on one side or the other” usages, the first few I looked at were definitely “on both sides”. For instance, in a 1796 book of magazine articles: “On either side of the sun, at north and south, about the distance of 45 degrees, was the segment of a rain-bow, inverted.”
Reply from Paul: While that’s interesting information, I’m going to continue to recommend common sense with the usage of either. Use the standard, modern usage of “on one side or the other” unless you prefer that the listener, not the speaker, decide the meaning of the sentence.
Your phrase “a night stand was placed on both sides of the bed” makes me picture a single nightstand arching over the bed to be on both sides. It would be better to say “a night stand was placed on each side of the bed” to indicate that each side has one, rather than that both sides are sharing one.
That aside, I completely agree that “on either side” is abused. It is, I think, a lazy writer’s way out of having to decide between “both” and “each”. I also make the argument that using “on either side” is wrong, because “either” is the dual of “any” and you would never talk about having “one on any side” of something that had more than two sides.
What else is weird is that it seems to happen only with the noun “side.” No one ever says things like, “Either of my pockets has a hole in it,” instead of “both my pockets have holes in them.” This suggests that “on either side” is idiomatic, and not based on a literal meaning of “either.”