Gemara

Eivus bar Ihi made a Lechi for the Mavoy in which he lived. Shmuel approved;

[After Shmuel died,] Rav Anan knocked down the Lechi. Eivus protested.

The case was, an overseer used to eat in his house and sleep in a Beis ha'Keneses [open to the Mavoy. He ceased to sleep there after Shmuel died];

Eivus held that one's residence is where he eats. He assumed that Shmuel did not permit due to the Beis ha'Keneses;

[Rav Anan knew that] really, Shmuel holds that residence is where one sleeps.

Shabbos 23a (Rav Yehudah): The first night, one who sees Ner Chanukah makes two Berachos. (He omits 'Lehadlik'.) On other nights, one who sees makes one Berachah.

(Rav Huna): If a house has two Pesachim (openings) to a Chatzer, Ner Chanukah must be placed in both.

(Rava): This is only if the openings are on different sides of the house.

We are concerned lest some people pass by only on this side, and suspect that just like the owner did not light on this side, he did not light on the other side!

Rishonim

Rambam (Hilchos Megilah and Chanukah 4:11): If they light for a guest in his house, they need not light for him where he lodges. If he has no house in which they light for him, he must light where he lodges. He joins with them in the oil. If he had his own house [where he lodges], even if they light for him in his house, he must light where he lodges, due to passersby.

Rashba (1:541): If one eats in his father-in-law's house, with his wife and children, even if he has his own Pesach, nowadays that we light in the house, and not outside, even if he has two openings, he need not light there. Everyone knows that we light inside.

Rashba (542): Also one who eats on his host's table, even if he sleeps by himself, need not light. In any case, he must join (pay for a share of the oil and wicks), or the host must be Makneh [transfer ownership] to him a share. This is unlike a guest for whom they light in his house. There, they already lit for him. Here, he must light, just like one who has no house elsewhere must join with a Perutah.

Mordechai (Shabbos 267): The Berachah for seeing Ner Chanukah is for one who will not light, e.g. a guest who did not hear the Berachah [from his host], and they light for him in his house. He is obligated to see. We say that the first night, one who sees makes two Berachos, and on other nights, he says one.

Poskim

Shulchan Aruch (OC 677:1): If they do not light for a guest in his house, he must give a Perutah to his host to be a partner in his oil of Ner Chanukah.

Magen Avraham (1): The host must add some oil due to the guest (Bach). The Rashba says that it suffices for the host to be Makneh some of his oil to the guest for a gift.

Kaf ha'Chayim (4): Eliyahu Rabah connotes that he must add oil to burn for another half-hour. Most disagree.

Mishnah Berurah (1): The same applies to a bachelor who learns outside his house, if he eats alone. If he eats on the host's table, he is like part of the household. Likewise, since the host regularly feeds his servants, letter of the law they need not light, unless they want to be Mehadrin (do the ideal Mitzvah, in which everyone in the household lights).

Mishnah Berurah (2): Letter of the law, it suffices if he is sure that his wife lit for him in his house. If he is unsure, he lights with a Berachah. Therefore, if one did not know that they lit for him, and comes home at night and found that they lit for him, he must light. Since he did not know from the beginning, presumably he intended to light for himself and not to be Yotzei through his wife, so Chachamim's enactment applies to him. The Magen Avraham holds that he lights without a Berachah.

Mishnah Berurah (3): If the guest has his own Ner, it is better to light for himself.

Mishnah Berurah (4): Since he does not eat here regularly, he is not like part of the household, and he must join with the host. The guest must hear the Berachos from the host.

Kaf ha'Chayim (2): One Perutah can acquire a share for all the nights of Chanukah. Or, the host can give a gift.

Kaf ha'Chayim (41): Gan ha'Melech (41) says that if he pays for his food, he must join or light for himself, but not if the host supplies all his needs. I.e. if he pays for the food and drink he consumes, he must join or light for himself, but not if he pays a fixed amount per day or week, or eats for free. It is better to explicitly ask the host to be Makneh a share of the oil.

Shulchan Aruch (ibid.): If he has his own opening, he must light there, even though he only sleeps there and he eats on the host's table.

Tur: A guest must light for himself. He was not Yotzei through the host. However, it suffices to give to him a Perutah and join with him. If they light for him in his house, he need not light at his Pesach. The Rosh wrote that if one eats at his father's or friend's house, and he has his own place to sleep, he must light. People see him go in and out, so if he does not light, there will be suspicion. People do not know that he eats elsewhere. We light inside. Presumably, the Bnei Chatzer know that he eats elsewhere. Even so, suspicion applies, for neighbors pass by Pesach ha'Bayis and see that he did not light.

Beis Yosef (DH Achsnai): It is clear from Shabbos 23a that when he has his own opening, he must light there, due to suspicion. The Mordechai obligates him to see a Ner Chanukah. The Ri says that people who go to fairs, and there are no Yisre'elim there, light in the Nochri's house.

Magen Avraham (3): Nowadays that we light inside, even guests light, but only if they eat in a separate room.

Magen Avraham (4): Since he lights only due to suspicion, he need not bless (675:25). One can distinguish from there.

Kaf ha'Chayim (9): The Shulchan Aruch connotes that even when he has his own Pesach, if his wife light for him at home, he is exempt. R. Yerucham says so. However, the Tur and Rambam explicitly say unlike this. The Shulchan Aruch was too concise. It is a Safek, so he should be Yotzei through another's Berachah, or bless without Shem and Malchus.

Mishnah Berurah (7): Some say that nowadays that we light inside, every guest must light for himself, even if his wife light for him in his house, due to suspicion. Even joining with a Perutah does not help, for the custom is that everyone lights in his house. The Magen Avraham is stringent like this only when he has his own room. Some say that when he relies on his household, when he sees Ner Chanukah he blesses she'Asah Nisim. Therefore, it is better to light for himself at least one Ner.

Kaf ha'Chayim (21): Some say not to bless on seeing. We are lenient about Safek Berachos. It is best to hear from the host.

Mishnah Berurah (8): If they do not light for him in his house, he must bless.

Shulchan Aruch (ibid.): The same applies to a son eating at his father's table.

Beis Yosef (DH Kasav): The Tur brings from the Rosh that if one eats at his father's or friend's house, and he has his own place to sleep, he must light. The Rashba exempts, but obligates joining with a Perutah, like a guest. Mahari Avuhav brings from Orchos Chayim that an adult son in his father's house who does not rely on his father's table is like a guest, and he must join. The same applies to a son-in-law, when he does not have a house in which they light for him.

Magen Avraham (5): Since it is due to suspicion, this is even if he regularly eats with his father.

Rema: Some say that nowadays that we light inside, he lights where he eats. This is the custom.

Taz (2): If one eats on his host's table, even if he sleeps by himself, the Rashba exempts from lighting, but obligates joining with the host, like a guest who has no house elsewhere. The Rema inferred that it does not depend on where he sleeps. If it did, he would need to light where he sleeps! It seems that also the Rema discusses only one who always has one special room for eating, and one special room for sleeping. The room for eating is primary, like the Rashba said. However, if one has a residence in the city, and once leaves his house and eats with a friend, surely he does not abandon his house and light in his friend's house. He must go home and light there. If his desire to eat seized him, and he does not want to budge from there, he should command his household to light for him in his house. In the house in which he is eating, he is only like a guest for whom they light in his house. He does not light where he is. I wrote above (Sa'if Katan 1) that a guest may light also where he is. This is only if he also sleeps there, but not if he merely eats there for an hour or two and returns to his place. There is no reason to light there, instead of in his house. This is like one who was in the street at the time to light. He cannot light there! Some people, when they eat elsewhere, send to bring their Neros [which they lit in their house, to burn where they eat], based on this Rema. They err.

Taz (2): The Maharshal says "if one eats in his father-in-law's house, if he can, even if he sleeps there, he should not join with him, rather, light for himself, to be Mehadrin. This is 'Ish u'Beiso.' If he sleeps in another house, obviously, he must light where he sleeps." This is unlike the Rashba. I say that the Rashba is primary. I learn from Eruvei Chatzeros (370:5). If one eats and sleeps in different places, where he eats is primary. The Tur brought from the Rosh that one must light where he sleeps. That was for the days of the Gemara, when there was concern for suspicion. We light inside, so only the Bnei Bayis see. We are not concerned for where he sleeps, for the Bnei Bayis know that he eats and lights elsewhere,

Magen Avraham (6) and Mishnah Berurah (11): Even the Rosh (the first opinion) agrees to this [like the Taz explained].

Magen Avraham (7): If one eats in another's house on occasion, he must light in his house, or his wife lights for him. It is a bigger Mitzvah to light himself than through his wife.

Mishnah Berurah (12): It is better to light for himself than to rely on a Shali'ach.

Bi'ur Halachah (DH b'Makom): If one regularly spends the entire Chanukah with his entire household in his father's or father-in-law's house, and eats and sleeps there, even if one day he eats at home, he lights where he eats at night and sleeps.