And what if you DO mean to say "Give the menu, please" ? Maybe you're asking someone to give the menu to someone else, not yourself.
For example, could you say "Дайте йому меню" ?
I think that the word "мені" is implied in this sentence, and that you'd have to put the pronoun in any other case. I can't be absolutely sure of this, though.
It's a typical Ukrainian phrase to ask for a menu in a restaurant or a cafe. "Me" (мені) is implied. If you meant "him," "her," ..., you would specify that.