Praise be to Allah.
It is permissible for a woman to say Ameen out loud, if she is in the presence of her mahrams and there are no “strangers” (non-mahram men) present, because a woman is forbidden to raise her voice in the presence of non-mahram men for fear of fitnah (temptation). But in the presence of her mahrams she is not forbidden to do so.
Shaykh al-Fawzaan (may Allaah preserve him) was asked about a woman speaking aloud during prayer, and he said:
It is mustahabb for her to recite out loud in prayer, whether it is fard (obligatory) of naafil (supererogatory) (i.e., the prayers in which Qur’aan is to be recited out loud), so long as no non-mahram man can hear her, lest he be tempted by her voice. But if she is in a place where no non-mahram man can hear her, then in prayers offered at night she should recite out loud, but if that will disturb others then she should not recite out loud. But in the case of the prayers that are offered during the day, she should recite silently, because the prayers during the day are to be recited silently. So it is not mustahabb to recite out loud in the prayers that are offered during the day, because that goes against the Sunnah. And Allaah knows best.