So, the answer to this will probably vary greatly depending on who you talk to and what their area of study is. There isn't one clear resounding answer as to whether or not Historians agree. Some accept only documented datable primary sources as accurate and consider all other evidence only useful in terms of the context of available evidence while others are far more trusting of written and oral traditions. In my "field" (Indian religions) much of what we have are oral traditions, or at least began as such. The language of many of these oral traditions is actually what we use to date when they came from and give us some idea of the time when they were written, but there is disagreement over the exact centuries and the reliability of the account. For instance, the Ṛg Veda probably began composition sometime in the 16th century and was codified around the tenth. This gives us some important contextual evidence for the society and culture of the early Indo-Aryans and their culture. We also can glean some political and other general historical knowledge from the hymns, but not much. The reliability of this is often called into question, but the analysis of the language and comparison with archeological finds tends to line up, if not exactly, with the hymns which were preserved through specialized memorization techniques and passed down word for word, even when Vedic Sanskrit was long out of fashion. That said, there are some hymns and verses that are challenged as later interpolations, such as the (in)famous "Hymn of Man" in the tenth mandala. So we need to take these hymns with a grain of salt and compare them internally and externally to get a proper picture.

For a second look, we turn to the Pāli canon, one of the main Buddhist canons. For around three to four centuries, the canon was entirely oral, and thus diverged from the original meaning to some extent. However, many assert that they are a fairly reliable picture of early Buddhism. We see this when we compare the Pāli Canon with other texts, such as the Chinese Āgamas, we see a picture start to emerge that gives us not only information about the practice of early Buddhism, but also the culture and politics of the 5th century Northeast India. All of this is disputed by some scholars, who assert that we can know nothing about Buddhism or Indian politics before the 5th century CE when we have the early extent texts, or at the very least before Aśoka. However, we do have other corroborating evidence, like inscriptions and reliefs.

As far as living oral traditions, the process is similar. Internal and external comparison, philological analysis, and "textual" analysis for consistency. How much stock we put in them depends on the narrative type and story. Some are obviously folklore or myth, but others have a grain of truth to them and are very old. We tend to say that living memory is around 100 years or so, but in actual fact, it varies significantly by culture and some cultures record events much farther back.