Yevamot 13 - A mitzvah marriage of Beit Hillel is illegitimate for Beit Shammai
Earlier we learned that if a man is forbidden to marry the wife of his deceased brother, because she is his close relative, all other co-wives of this brother are equally forbidden to the man, and no halitzah or yibum is required.
Beit Shammai disagree with this basic rule and permit one to marry a co-wife of the deceased brother. This leads to many important consequences, for example, if co-wives performed a halitzah, they are like divorced and cannot marry a Kohen - that is according to Beit Shammai, but for Beit Hillel, who say that the halitzah was not needed in the first place and means nothing even if performed - they can still marry a Kohen.
More importantly, if a brother of the deceased makes a yibum to his co-wife and marries her, following the opinion of Beit Shammai that this allowed, the children of such marriage will be mamzerim according to Beit Hillel, and mamzerim are not allowed to marry a Jew at all, and the child of a mamzer is still a mamzer!
Even though they had such dividing disagreements, Beit Shammai and Beit Hillel still maintained a friendly relationship, would intermarry, and relied on each other for purity laws of sacrifices. This is because they trusted each other, so that if a situation that is permitted according to one but forbidden to another arose, they knew that they would be honestly informed.
Art: Group portrait of a husband and wife in a drawing room by Arthur Alfred Davis