In general, Sepharadim follow the opinion of the Shulhan Aruch (Orah Haim 671:2), which says that regardless of the number of members in a family, the head of the household lights one Menora on behalf of everyone. The Ashkenazim follow the opinion of the Rama (ibid.), who says that each member of the household light one’s own Menora.
There is a question as to whether a household member could opt out of fulfilling the Mitzvah via the head of the household and perform it oneself. This may be applicable if a household member arrives home later than the family lighting time, for example, and wishes to light oneself. Based on the Shulhan Aruch’s ruling, many later rabbis ruled that this would not be permissible since every household member’s obligation is automatically fulfilled when the head of the household lights. Furthermore, they explain that one who does light one’s own Menora who be reciting an unnecessary blessing. and that doing so would constitute an unnecessary blessing.
Nevertheless, Rabbi Shalom Messas (Tevuot Shemesh, Orah Haim § 7 & Shemesh Umagen vol. 2, § 3) says that the reason the Shulhan Aruch rules against lighting multiple Menorot is only because it would not be evident to a passerby or an onlooker what day of Hanukkah it was. For example, on the first night of Hanukkah, if five family members light five individual lights, it may appear to an onlooker that it is the fifth day of Hanukkah. Rabbi Shalom Messas responds by saying that if the Menorot are placed sufficiently apart from each other, then it is clear even to an onlooker that these are separate lights. Regarding the issue of an unnecessary blessing, he references the law of Birkot HaShahar (Orah Haim 6:4) which states that even though one person is reciting the morning blessings aloud in the synagogue and everyone answers “Amen”, each person may recite one’s own blessings. Furthermore, he states that the issue of an unnecessary blessing would only apply to one’s wife since husband and wife are considered to be one unit vis a vis the candle lighting.
Summary: One who normally would be acquitted by the head of the household may nevertheless light one’s own Menora, as long as one explicitly intends that one’s obligation not by fulfilled by the head of the household.