Does the word, ‘napkin’ have a metaphoric or symbolic meaning other than a piece of cloth or paper used at meals?
Solution 1:
There's nothing special about napkin here; I believe this is a play off of either side of the aisle, which is an expression used to refer to the two political parties (who often sit on opposing sides of the aisle in the halls of congress).
In this case, I get the feeling that either side of the napkin refers to the steakhouse employees (who work on one "side of the napkin") and the customers (who dine on the other). In the article, the steakhouse owner chimed in with one theory; in subsequent paragraphs, some politicians offered explanations as well.