I think one thing that is important to note is that the relationships between Native Americans and Europeans were really, really, REALLY complex. One dynamic we see time and time again is trade, both sides benefited from trade and at the same time were always looking for any advantages. If you have connections to exotic goods you have power, we can see this in the social structures at Cahokia in 1100 AD as well as in the 17th century. This isn't unique to any one culture, but I did want to note it as many stereotypes on Native Americans seem to paint these communities as being fairly altruistic and almost naive on matters of trade. While altruism did occur, Europeans were often accepted into communities due to the potential to utilize the connections they had. You mention the French in another comment, and they are a really good example of this. In 'Archaeology of the Soul', one of my personal favorite texts, Hall describes a situation where French colonists develop a community close to an existing tribal community. To sum it up, at first the colonists are allowed to live in the area and are not harassed. Remember, these colonists would have brought with them trade goods like worked metal utensils/tools. I had a professor who once told me that the best way to date copper artifacts was to see if it was a modified pot as once Europeans arrive it was easier to trade for pots and dismantle them than it would be to find copper to work into tools. More anecdotal than anything, but it highlights how sought after these trade goods would be.

But back to the colonists, over time as they began to attract more colonists and bring in livestock the local tribal leaders began to become concerned. The power dynamic was shifting and the colonists were quickly shifting from a resource to competition. So they raided the community and ended up briefly chasing the French out which was the intent of the raid, to show strength and power over this outside group.

On the other hand, another power dynamic would be the European trader living amongst a community. Often these Europeans would be adopted into the community, assuming the name of a departed member of a powerful lineage. Again this is a means to build power, this adoption would connect the prestige goods with the leader's lineage. On a somewhat related note, certain communities would position attractive women near French traders in the hopes of 'swaying' these male travellers who would then possibly be a bit easier to strike a deal with. There is a reason why the Grand Tetons were given that name, being a single guy roaming a foreign land makes you miss some aspects of home more than others.

But in all seriousness I think the best way to describe how the Native tribes viewed Europeans would be that most saw them as just another community. For centuries trade and its associated power struggles dominated how tribes interacted, so when a new group appeared they fit well into the exiting structure. Outsiders were accepted in more readily if they served to benefit those in power, if they became a threat they would be dealt with to maintain the power structures in place.

Is there a specific region/time period you have in mind for this? The colonizing nations would have not only had different approaches to the Native populations, but also changed this approach depending on the situation at hand. Add to this changing leadership on both sides and what was allowed in 1703 might not be the case in 1789.

Edit: better phrasing on my part would be that both the Europeans and Native Americans would have had different motivations at various points. Did not mean to take away agency from the native populations, my mistake

The challenge with your question is that most of the writing systems of pre-European contact and early contact Americas haven't been deciphered. The one we're the most aware of is Classic Maya, and its inscriptions weren't deciphered until the mid-twentieth century.

Since most information recording systems of the indigenous peoples have yet to be deciphered, we don't know what was recorded. It could have been literature and history, it could have been census data, it could have been recordings of religious omens. Birch bark scrolls are known throughout the world, including the early Americas. Without translation, they are indicative of many things, but not indicative of their perspective on their history.

Much of the history passed down by the indigenous Americans has thus been oral history, but that has its own challenges — first, while we know that disease made "inroads on" or "affected" local populations from American history classes as a child, we're not aware of what it was. It was a catastrophic event. Charles Mann, in 1491, suggests that the death rate could have been up to ninety-five percent. Unsurprisingly, that's going to hollow out any traditions of history keeping by tribal elders, to state it mildly. Can't have an oral tradition if there's no one left.

The other issue is that where oral traditions survived, they focused on issues far more important to the tribes and communities than whether newcomers were being adopted. Most of the histories we have found instead discuss more fundamental social issues, such as mythology. This is valuable, but not for your purposes.

The closest well-verified example I can give from my own studies is John Demos' The Unredeemed Captive, published 1994. Eunice Williams was one of the captives from the Deerfield settlement, taken by Mohawk warriors in a joint raid with the French in 1704, when she was seven years old. Her surviving neighbours went home after three years of complicated negotiations; she did not. She'd been taken into a Mohawk home and reared as an adopted child by a recently bereaved mother. By the time she was ten and able to return home, she was a baptised Catholic who spoke the local languages fluently.

The French negotiators involved refused to intervene, since she was adopted by the Mohawk. The other Williams children returned to their Puritan settlement.

Eunice went on to marry a local Mohawk man at the age of 16, and live until an astonishing 95 years of age. She visited her New England relatives several times with her husband and children, but did not recant or resettle in New England.

The problems with captivity narratives (and there are a few captivity narratives out there) is that they were most likely to find publication if they held up to the common narrative of life with the native tribes as being confusing and frightening, rather than life with the native tribes being appealing. Moreover, due to the nature of them, most of them were written by persons who weren't candidates for adoption into the tribe, because they were full adults. People who were adopted into the tribe and chose to stay were removed from the European publication system as they no longer identified as Europeans.

One of the issues here is that the way that people had ongoing contact with Natives was either for business purposes of trade, negotiations in land settlement, or proselytisation of Christianity; otherwise, the contacts would be hostile and sporadic. The vision of the era was that tribes were attacking out of nowhere in superior numbers completely unprovoked — however, from the modern perspective, we know this to be completely unlikely. Attacks are expensive and run the risk of casualties, and use weaponry that could be used for hunting instead of fighting outsiders. For the community to decide that it was a good use of their resources, something had already happened, whether it was the encroachment of settlements on lands, attacks by Europeans, or alliances with other groups who had reasons for their hostilities.

Even if there were peaceful instances of cooperation — and there definitely were some in French colonies, and occasional instances in English colonies — that peacefulness would go out the window if a daughter or widow decided to move into a native community by preference and take a native husband.