Translation:Both the menu and the plate are on the table.
Because of the initial "и" - Russian "и ... и ..." equates to English "both ... and ..." It's a perfectly valid sentence in English without "both", just as it is in Russian without the first "и", but if you leave it out your translation doesn't correspond to the Russian sentence anymore.
I think - and I am just making an educated guess here - is that "menu and plate" are collective nouns, suggesting that there is more than one menu and more than one plate on the table. The Russian is singular, signifying that there is just one menu and one plate. Your sentence is thus ambiguous as to the number of items on the table.
Apparently it actually is necessary, if the point made above by Theron126 is correct. It's not bad English to leave it out, but it seems that the И..., И... construction in Russian actually means "both". I assume that the Russian would have to be меню и тарелка if you wanted to translated it without the "both".
I was trying to be clever and wrote "Both the menu and the plate are on the desk", since стол is both table and desk (?). I was marked incorrect, however.
Is this just a contextual thing? As in, is it just that you're not likely to find a menu and a plate on a desk, but you would on a table? Or is there another reason why the translation is incorrect?
I was marked wrong for 'Both menu and plate are on the table'. And yet that's exactly how I would say it conversationally. You don't need all the extra articles to indicate quantity - it just sounds stilted that way. The fact that I said menu and plate rather than menus and plates is already enough information. Boo Duo!
"Both menu and plate are on the table" is a possible sentence, but wow, that is a hyper-specific construction, and I cannot even begin to articulate how and when that structure might be used. My immediate impression is that that structure exists in poetry and much older English. It's possible it's a feature of Englishes other than American. Are you a native speaker, and, if so, where are you from? To me, not using an article here is, as I said, technically possible, but only something I've come across in literature (American English speaker)