I'm working on an outline for a short presentation on how new online tools, especially this site, interact with the traditional Jewish learning models of chavruta and teacher-student. What are the main source texts (I suppose mostly in the Talmud, though some may be earlier and later) for the roles and benefits of these two models and the interplay between them? If there's a particular commentary or contemporary source that sums up traditional understandings of any of these primary sources, that would be great to have, too.
A few raw blobs off the top of my head.
The Gemara has "Cherev al haBadim" -- a curse on those who study all alone. Moreover, they become foolish and make mistakes. I (Shalom) heard one rosh yeshiva say that means you always need a chavrusa ("and I even asked my chavrusa to come to my house during sheva brachos week so we could learn"); and another who said that means you can study alone, but every so often compare with others ("some people even send a chavrusa to the house for sheva brachos! What kind of nonsense?").
Choni Ham'agel's "oh chevrusa o misusa."
Shaul Stampfer's work on the 19th-century yeshiva says it appears in Volozhin (at least at one point), a chavrusa would be for a weaker student if he needed help; most studied alone.
Gemara in Sanhedrin that in Babylonia "mechablim zeh et zeh bahalacha", ("we beat each other up arguing over halacha"), whereas in Israel "man'imim zeh et zeh" ("we support each other pleasantly." I once witnessed this with two Israelis, one going, "I'll take what you're saying one step further", and the other "I'll take what you're saying one step further", even if that wasn't what was intended!)
I don't know if this is what you are looking for, but the famous story surrounding the death of Reish Lakish (bottom of Bava Metzia 84a) plays out the benefit of a proper chavrusaship. A translation is here.
R. Yochanan was upset following Reish Lakish's death, and the rabbanan sent R. Elazar b. Pedas to learn in Reish Lakish's stead. R. Yochanan felt the chavrusaship was unproductive, since R. Elazar merely supported (textually) everything R. Yochanan said. With Reish Lakish, R. Yochanan had to defend his position and this allowed a broader understanding of the halacha.
Mishna 5 Perek 5 in Sanhedrin mentions a partnership in learning between dayanim in the case of a death sentence that is unfavourable. In such a case the final judgement is delayed until the following day. Part of the mishna mentioned above reads:
אם מצאו לו זכות, פטרוהו. ואם לאו--מעבירין דינו למחר, ומזדווגין זוגות זוגות, וממעטין במאכל, ולא היו שותין יין כל היום, ונושאין ונותנין בדבר כל הלילה, ולמחרת משכימין ובאין לבית דין.
This loosely translates as: "if they find him meritorious he is exempt [from punishment]. If not, his sentence is delayed until tomorrow. [The dayanim then] pair off ['...in order to deepen their understanding' (Kehati)], they diminish in their food intake, don't drink wine all day and critically analyse the case all night ['each and every dayan in his own pair, or alone' (Kehati)]. The next day they arise to the Beth Din".
Although its not possible to learn out a general rule for chavruta study from this mishna, it can certainly be seen as meritorious and something that was done for in-depth understanding.
A source for the role and value of havruta-style learning is Taanit 7a which states
אמר רבי חמא (אמר רבי) חנינא מאי דכתיב (משלי כז, יז) ברזל בברזל יחד לומר לך מה ברזל זה אחד מחדד את חבירו אף שני תלמידי חכמים מחדדין זה את זה בהלכה
Rabbi Ḥama, son of Rabbi Ḥanina, said: What is the meaning of that which is written: “Iron sharpens iron, so a man sharpens the countenance of his friend” (Proverbs 27:17)? This verse comes to tell you that just as with these iron implements, one sharpens the other when they are rubbed against each other, so too, when Torah scholars study together, they sharpen one another in halakha.
an additional resource on the efficacy of this mode of study is the work A Philosophy of Havruta: Understanding and Teaching the Art of Text Study in Pairs
Regarding the teacher-student learning model, it was not clear to me from the question if this refers to pedagogy or the student serving/attending to the teacher. For the latter please see שימוש חכמים and in particular Berachot 7b
וא"ר יוחנן משום רבי שמעון בן יוחי גדולה שמושה של תורה יותר מלמודה
And Rabbi Yoḥanan said in the name of Rabbi Shimon ben Yoḥai: Service of Torah is greater than its study,
i.e., serving a Torah scholar and spending time in his company is greater than learning Torah from him.(translation and commentary from Sefaria)
See also ibid 47b and Avodah Zara 19a
Though there are examples in the talmud of the teacher asking questions to 'sharpen' the students; my own analysis is that the teacher was seen as the source for information whereas the study partner was used to analyze the plumb the depths of that information.