See further in the sources, below.
The Shulachan Aruch writes (695:4) that both men and women are obligated in giving matanos la-evyonim (Shulchan Aruch 695:4; see Peri Chadash, who writes that women are not obligated).
This statement implies that even married women are obligated. However, the Magen Avraham (694:13) writes (concerning mishloach manos) that “I have not seen people being careful in this matter. It is possible that the ruling applies only to a widow, and not to a married woman, for whom the husband sends to several people.” The Magen Avraham concludes that one should be stringent, and a married woman should send her ‘own’ manos, rather than relying on her husband’s sending on her behalf.
The Aruch Hashulchan (695:18) likewise writes that a woman is obligated in all the mitzvos of Purim, stating further that even if she is married, she is not exempted by her husband’s giving. However, he writes elsewhere (694:2) that “a man and his wife fulfill their mitzvah with a single gift, for they are as a single body.” It is possible to resolve the seeming contradiction by distinguishing between a husband who gives on behalf of his wife, and informs her of this (by which the wife performs the mitzvah), and a husband who gives without specific intention for his wife (with which a woman does not perform the mitzvah).
Authorities add that even if a married woman is obligated to give matanos of her own, her husband can give the matanos on her behalf, and there is no need for the wife to actually take prior possession of the money.
The reason for this is that with regard to sacrifices, we find that a man can bring his wife’s sacrifices (seeRambam, Shegagos 10:6). We thus derive that the same principle applies to gifts to the poor (see Iggeres Ha-Purim, Chap. 4, quoting from Rav Nissim Karelitz;Shaarei Yemei Ha-Purim p. 130).
Concerning children over the age of bar-mitzvah, who are supported by their father, the Magen Avraham (694:12) writes that there is no obligation of giving matanos—because children have no possessions of their own. He adds that nevertheless, one should be stringent in the matter, and instruct children to give. A similar principle is found in the Leket Yosher (customs of the Terumas Hadeshen), who records that the Terumas Hadeshen would train his children in sending mishloach manos, estimating that the children were “at most fifteen years old.” Unlike a married woman, a person should transfer possession of the money to his children, so that they can perform the mitzvah with their own money.