This past weekend was the Synagogue Council of Massachusetts annual Unity Shabbaton. One of the scholars-in-residence talked about his idea that many of the discussions in Mishna Eduyyot are trying to deal with the aftermath of zealotry on the Jewish community of the second century. Eduyyot consists entirely of rulings on a wide variety of topics where the source of those rulings is 'thus and so directly learned the tradition from Great Rabbi X'. Some historians think it was the first part of the Mishna written down.
The Mishna whose interpretation I found problematic (In Eduyyot 4:8) discusses how Beit Hillel and Beit Shammai had a number of differences of opinion which impacted personal status and ritual purity. These differences were consequential. It mean that there were people who Beit Hillel thought could marry a Cohen, or even marry a regular Jew, who Beit Shammai thought were ineligible and vice versa. The dispute also extended to whether certain utensils were ritually pure or not. Widespread disagreement on this could have theoretically meant that members of Beit Hillel could not have eaten in the houses of followers of Beit Shammai and vice versa.
However, the Mishna concludes "And although these pronounce unfit and these pronounce fit, Beth Shammai did not refrain from marrying women from [the daughters of] Beth Hillel, nor did Beth Hillel refrain from marrying women from [the daughters of] Beth Shammai." (and similarly reagarding utensils).
When I learned this, what I was taught it meant was that if a member of Beit Hillel wanted to marry a daughter of Beit Shammai, he might be told "Although Beit Shammai holds that this woman is fit to marry, by the rules of Beit Hillel she is not." and vice versa. The teacher told me this was the view of the medieval rabbi Ovadiah Mibartenura.
The teacher's view, which I have previously seen ascribed to Judith Hauptman (Masoret Magazine v7n3), is that this meant that Beit Hillel and Beit Shammai each accepted the other's definition. Thus if my daughter is acceptable for marriage according to Beit Hillel, a man from Beit Shammai would be willing to marry her.
I have a very hard time accepting this interpretation. There are two ways to look at it, and for each I will provide a source for refuting it. One interpretation is that Beit Hillel and Beit Shammai each said "Well, we'll accept the other's ruling." This obviously doesn't work. Now the man from Beit Shammai would say "I will marry you since you are acceptable according to Beit Hillel, but the daughter of Beit Hillel would say 'but I must refuse because I am forbidden to you according to the rules of Beit Shammai." This provides no gain in what is actually permitted and has the further problem that each person think their own decision is the wrong one. This scenario makes me think of the saying in Pirke Avot "One who says - What's mine is yours and what's yours is mine - this is an ignorant man."
The other interpretation is that if either Beit Hillel or Beit Shammai ruled that a woman was fit for marriage then both groups accepted the ruling. But in the Talmud (Chullin 43b - 44a) the Gemara quotes a Beraisa that states, "One who follows the lenient rulings of Beis Shamai and the lenient rulings of Beis Hillel is a Rasha. One who follows the stringent rulings of Beis Shamai and the stringent rulings of Beis Hillel -- of him the verse says, 'The fool walks in darkness' (Koheles 2:14). Rather, one must follow either Beis Shamai consistently, both his lenient and stringent rulings, or Beis Hillel consistently, both his lenient and stringent rulings."
This conflict has consequences in the present as well. The overwhelming majority of Orthodox Jews will not accept the validity of the overwhelming majority of conversion conducted under Conservative Jewish auspices. I know people who reason as follows "I follow the halacha as understood by the Conservative movement. By these rules my conversion is valid and I am a Jew. Therefore, when I enter an Orthodox synagogue I can simply tell them I am Jewish, and be counted towards the minyan, lead services if asked, etc." To me, the Mishna in Eduyyot should cause these people to say "Even though I am a Jew according to my rabbis, I am not according to yours and therefore you should not count me towards the minyan."
UPDATE: Rabbi Hauptman's article can be found on the Internet Archive
UPDATE 2: Reading other people's commentary on the article, Meredith Warshaw brings out the point that while a member of Beit Hillel would not marry a woman of Beit Shammai who was prohibited according to BH's understanding, he would marry a woman of Beit Shammai who had no obvious defect. This isn't inevitable - one could imagine someone saying 'Perhaps 4 generations ago this woman's ancestor was a mamzer according to me, but not according to Beit Shammai. How can I take the risk?'