I owe thanks to Anonymous, G, and R' Gil Student for recommending that I read The Dynamics of Dispute: The Makings of Machlokess in Talmudic Times by Rabbi Zvi Lampel. The book has proved invaluable to me.
CORRECTION: Contrary to what I had thought, in fact the book IS in print, in a brand new edition! It is available from Judaica Press, or should be available in your local Sefarim store.
There is one particular chapter that I think should be xeroxed and handed out to every single Jewish teacher in every elementary school, high school or other place of learning. This is the chapter on the meaning of "Eilu V'Eilu," namely the adage that "These and These Are The Words of the Living God." It is Chapter 11. Having had this fight as recently as last year with a teacher I otherwise highly esteem, this chapter beautifully answered my questions as to the ways in which this adage can be understood, and the correct (and incorrect) application thereof.
Lampel includes a review section which reviews the differing opinions of leading figures as to how to understand the meaning of "These and These." I have reproduced it below, although one really ought and must read the entire chapter to understand how these viewpoints are formulated (he carefully provides both the Hebrew text and English translation in order to demonstrate how these points of view are derived).
The following is a brief summary of the explanations we have quoted for the meaning of "These and those are the words of the Living God."
Rav Yisroel Salanter: All valid attempts to reconstruct what Hashem told Moses are subsumed under the title of "Torah," including the opinions that are ultimately nullified as halacha.
The Yahm Shel Shlomo: Since a Sage's conclusions conform to sound logic, they are as valid and holy as the explicit words of Hashem. The kabballistic teaching that all souls were present at Mount Sinai and each perceived the Torah through one of forty-nine conduits supports this position.
Ritva (Eurvin 13a) and Tosefos Shantz (Aid'yos 1:5) cite another kabballistic teaching that Moses was shown forty-nine arguments to each side of an issue (totaling ninety-eight arguments to each issue) and was told that the decision was entrusted to the future Sages. We have suggested that we may here apply Rashi's understanding (Menachos 29b) that this happened prior to Moses' receiving the Torah. Thus, "These and those" positions conform to human logic and even the Will of Hashem before He prescribed the limitations and parameters of halacha.
Rashi according to our first understanding: Perceptions are subjective. Trust the perception of present reality determined by your authority, though another's perception may be different, because it is a true "reality."
Rashi according to our second understanding: The situations the Sages are ruling on are not really identical, and all authorities would agree on how to rule in any given time. Disputes across generations are only apparently disputes.
The Ohr Gedaliahu: There is a single unifying formula which really produces opposite results in different circumstances.
Drashos HaRan: Each opinion is potentially the official halacha, and the decided halacha is the one we must obey- whether it is true or false- because chances are that the decisions do conform to the "original intent" and the benefits we accrue by obeying the Sages outweigh and counteract the risk of harm.
Tosefos: Regarding opposite reports of past occurrences, there are elements even of Absolute Truth to each side of a machlokess, though one is dominant and the others recessive (the Maharal of Prague's concept of "They were all given by one Shepherd"). Whereas one version reports a tradition describing the actual event, the other reports a tradition of a strongly considered action.
Maharal of Prague: Two statuses are actually equally present. Neither one is recessive, neither one is dominant. This was true only of some machlokos, the first of which were those of Bes Shammai and Bes Hillel.
Rashi according to our third understanding: All the criteria and considerations introduced by disputing Sages play a role in determining absolute truth, although in each individual situation the aplicability of these factors changes, thereby changing the situation's status in terms of absolute truth. (However, such unstable and subtle considerations are too complex and cumbersome to be allowed in practical halachic decision-making, and opposite reports about the proper halacha and the proper criteria for determining it cannot both be correct.)
This review of how our classical commentators understood the adage "Eilu v'Eilu Divrei Elokim Chayim Heim, These and those are the words of the Living God," showed that there are different ways in which it could be understood. But we can state unequivically that according to all, neither Hashem when He originally established the halacha nor Moses when he transmitted it, stated more than one halacha for a given case. So far as the halacha l'maaseh, the practical law that we should follow is concerned, two or more opposing Sages cannot both be stating what their teacher, or teacher's teacher, or Moses or Hashem originally said.
If Hashem told Moses that the law is one way, someone saying otherwise simply does not conform to that law. In this aspect, we cannot say that he is "right." "These and those" does not mean that. Nor does it mean that two Sages disagreeing over the meaning of another Sage's statements are both conforming to his actual intention. If their opinions are mutually exclusive, then that just cannot be. And we cannot imagine that two students could be both correct if they disagree over what their teacher's very words were. The opposite statements could not have issued verbally from their teacher's mouth at the same time. In all such cases, someone must be mistaken.
~The Dynamics of Dispute, Chapter 11, pages 224-226