Since fasting is not obligatory (farḍ) for a woman in a state of menstruation, there is no Sharʿī prohibition against her eating or drinking, whether openly or in private.
Nevertheless, in order to remain above suspicion in the eyes of those who may be unaware of her condition, and out of reverence for the sanctity of fasting and respect for those believers who are observing it, it is more befitting—according to the dictates of proper Islamic etiquette (adab)—that she eat and drink discreetly.