In the case of Norse mythology we are extremely blessed and cursed with the existence of the sagas which contain almost the entirety of our knowledge of Norse mythology. There are other sources from other cultural contexts as well, and the role of archaeology is increasingly gaining prominence, but in terms of "home grown" sources, the sagas are the overwhelming majority of what we have available to us.

Before moving on it is important to understand something. The sagas themselves were not the codification and primordial Germanic belief, unchanging through the centuries diligently written down by objective observers. They were written for a reason, in many cases to glorify particular families, and did not really have cross cultural knowledge to draw on. They are overwhelmingly concerned with the deeds of Icelandic and Norwegian men (and their off shoots in Greenland and Vinland). Yet they remain our best glimpse into Norse cosmology, mythology, philosophy, all those ologies.

A great many of the sagas were written by prominent and notable Icelandic leaders, they did not arise relatively anonymously from monasteries or by newly literate practitioners of the older religious traditions of the Norse. They came from Christian writers writing in a Christianized society and no small number of parallels and influences have been drawn between Christianity and figures and events in the Sagas.

However, more pertinent to your question, a great many of the sagas are attributed to a single individual, Snorri Sturluson. Now it is important to note that he did not write ALL of the sagas, and in some cases the attestations are tenuous, and some of have been connected to earlier individuals and now lost versions of these stories that existed before Snorri's versions that have survived to the modern day. There is also some debate over which stories he himself penned and which he simply compiled or edited.

The Prose Edda and the Heimskringla are attributed to him and both works contain elements of what is now considered "Norse mythology" as understood by most people. Ragnarok, Valkyries, Valhalla, and so on are all mentioned in various parts of both of these works, as well as genealogical information (and claims), and so on.

I cannot really speak to Greek mythology though.