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Consider these two sentences.

1. You will find delicious food in either New York or Los Angeles.
2. You will find delicious food either in New York or in Los Angeles.

My questions are as follows:
A. Are both sentences grammatically correct?
B. Do the two sentences say two different things, or do they communicate the exact same message? If they do say different things, what do they say? How do you know that they say different things?
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They say the same and are both grammatically fine. Simplicity of style recommends #1.
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Do they really say the same thing? I am tempted to say that number 1 claims "Only one of the two cities mentioned has delicious food." However, I think number 2 says "Delicious food exists in both of the mentioned cities." Is this interpretation incorrect? If so, why?
AnonymousI am tempted to say that number 1 claims "Only one of the two cities mentioned has delicious food." However, I think number 2 says "Delicious food exists in both of the mentioned cities."
Why so?
Since they contain two equal constructions, the meanings are the same. Either is used before the first of two (or occasionally more) alternatives that are being specified (the other being introduced by "or").
Yes, they both, I repeat, say the same thing-- that both LA and NY have good food, whichever one of them you go to. The meaningsob both, however, could be ambiguous for the very naive, who might think that one of these great cities lacks decent cuisine.
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Yes, both are grammatically correct, and though the first sentence might seem odd to the ear, there is no grammatical error in it.

And to the second question, yes, they do communicate the exact same meaning.

The only thing is that the proposition's (in) place / position has changed.

You will also find that the first form (in either NY or LA) is used less professionally.