Praise be to Allah.
Is one sheep sufficient for one man and his family?
One sheep is sufficient as udhiyah for one man and his family and whoever he wants among the Muslims. ‘Aishah (may Allah be pleased with her) said that the Prophet (peace and blessings of Allah be upon him) ordered that a ram with black legs, black belly and black (circles) around the eyes should be brought to him, so that he could sacrifice it. He said to ‘Aishah: “Give me the knife,” and she did that. He took it, then he took the ram, placed it on the ground and then slaughtered it (i.e., prepared to slaughter it), saying: “In the name of Allah, O Allah, accept (this sacrifice) on behalf of Muhammad and the family of Muhammad and the ummah of Muhammad.” Then he sacrificed it. (Narrated by Muslim)
Abu Ayyub al-Ansari (may Allah be pleased with him) said: “At the time of the Prophet (peace and blessings of Allah be upon him), a man would sacrifice a sheep on behalf of himself and his household, and they would eat some and give some to others.” (Narrated by Ibn Majah and by al-Tirmidhi, who classed it as sahih. Also classed as sahih by al-Albani in Sahih al-Tirmidhi, 1216)
If a man sacrifices a single sheep or goat on behalf of himself and his family, that will suffice for everyone he intended of his family, whether living or deceased . If he did not intend anything specific then it includes all those who are included in this word (family or household) either customarily or linguistically.
Customarily, it refers to all those whom he supports of wives, children and relatives. Linguistically, it includes all those who are related to him of his own children and the descendents of his father, grandfather and great-grandfather.
One-seventh of a camel or cow is equivalent to one sheep. If a man sacrifices one-seventh of a camel or cow on behalf of himself and his family, that is sufficient. Prophet Muhammad (peace and blessings of Allah be upon him) said that one-seventh of a camel or cow takes the place of a sheep in the case of the hadiy (sacrifice offered during Hajj). So, this applies also to the udhiyah because there is no difference between the udhiyah and the hadiy in this regard.
Can people share in a sheep?
If two or more people buy a sheep and sacrifice it, that is not sufficient, because no such thing has been narrated in the Quran and Sunnah.
Can 8 people share in a camel or a cow?
Similarly, if eight or more people share one camel or one cow, that is not sufficient (but it is permissible for seven people to share a camel or cow). This is because acts of worship are as prescribed in the Quran and Sunnah and are not subject to personal opinion. It is not permissible to go beyond the set limits with regard to how much is to be done or the way in which it is to be done. This does not have to do with including others in the reward, because it was narrated that there is no limit to the number of people on whose behalf the sacrifice may be offered.
And Allah knows best.