It would depend whether the other country required evidence of giving up your citizenship. most countries just ask you to "declare" that you are doing so. If the other country was really strict, then I guess they wouldn't grant citizenship.
If the person gets the second nationality, I think the person holds both nationalities, but the countries involved only recognise their own nationality.
so country A says that person is A-ish, and country B says that person is B-ish. If they are in country A, then they are effectively A-ish, if they are in country B, they are effectively B-ish, and if they are abroad, then they can pretty much use which ever citizenship is most convenient.