I was hoping somebody with more expertise on this topic in particular would come and answer, but since it's been a few days, I can try to cover it. I'm adapting some parts from this answer that I wrote to a different question.
Russian settlement in Siberia generally followed a basic pattern, beginning in the late 16th century and continuing until the end of the 17th. A band of Russians, often Cossacks employed by a noble family, would sail down a river, establish a fort where they saw fit, raid nearby settlements, take hostages, and then demand tribute, usually in the form of furs, from the local peoples. The word the Russians used for that tribute was yasak, and translating it as tribute already biases the discussion a little. You can also translate it as "tax" too — but it's hard to translate, because what relationship it represented is at the heart of the confusion over when and how Siberian peoples became Russian subjects.
For decades, yasak remained the most obvious Russian incursion into daily life in Siberia. Russians thought that their Siberian subjects were paying them yasak as a sign of fealty. What that actually meant in practice was that Siberian peoples would bring local Russian officials or Cossack military leaders rich gifts of furs or other goods, and be greeted with a feast and possibly other gifts in return, which was called the "sovereign's compensation." So is that an equal delegation bringing gifts as a sign of good faith and being given the gift of a feast in return, or is it a subject people paying tribute? The Siberian peoples seem to have thought it was an equal trade relationship, and only the Russians thought they were making subjects.
That does not mean that the relationship was all friendly, though. Those hostage raids I mentioned above caused a fair degree of displacement and hardship, and resulted in a pressure to pay the tribute rather than undergo more raids. It also made the spread of disease like smallpox easier. However, even when military force was used to "pacify" Siberian peoples, it's hard to tell exactly how each side perceived it. The "conquests" were finalized with what the Russians called a shert', a borrowed Turkic word they used to mean "peace treaty". But the word was alien to the Siberian peoples the Russians made peace with too, and they often interpreted it to mean a mutual pact. This was all the more so because already-subdued Siberians, "yasak people," were often convinced to or themselves offered to help defeat their rivals among the "non-yasak people."
Meanwhile, Russian peasant settlement followed the collection of yasak. At first, the Russians had tried to force some of their fur providers (who, yes, did settle temporarily but were often semi-nomadic pastoralists) to settle into agriculture, because transporting grain to Siberian Cossack forts from Europe was prohibitively difficult. However, this was unpopular among the Siberians and counterproductive for the Russians. The policy just turned skilled Siberian hunters into unskilled farmers who no longer provided furs. The state abandoned the attempt, much more out of its desire for furs than out of compassion.
Instead, then, it encouraged Russians to follow the Cossack raiders and work the land, mainly by offering them life as a free peasant. Serfdom was just becoming entrenched in European Russia at this time in the late 16th century, but even before then, peasants were subject to degrees of unfreedom, and even after, it was never instituted in Siberia. This, in practical terms, meant that there were two administrative systems patchworking across Russian Siberia: one of tax-paying, Orthodox Russian peasants clustered in villages and forts, and one of tribute-paying, "pacified," shamanist Siberian peoples. (There were also the "non-yasak people," but they were beyond the scope of your question and my knowledge.)
Russian administration, for the peasant settlers, was not much different from that in European Russia. Other than the fact that they were not enserfed in Siberia, they were still subject to the same taxes, judicial system, and forms of local village governance as European Russians — if you can call anywhere in the Russian Empire subject to the same taxes as anywhere else. It's pre-revolutionary France-level confusing.
Siberian peoples, on the other hand, experienced actual day-to-day governance primarily through yasak collection and through the upheaval caused by the Russian peasants' arrival, and not much else, unless they converted and went to live as peasants. Peasant settlement meant clearing large tracts of forest and hunting, fishing and trapping in much of the rest, which interfered with the lives of the local Siberians. This interrupted the fur supply from Siberian pastoralists, and the Russians became very cautious to make sure that they only granted peasants unused lands. But for the Siberians, it was much more direct a shock. It contributed to that feedback loop of displacement, hunger, and disease that, compounded by slavery, reduced their numbers significantly.
Meanwhile, the actual process of yasak collection, even though it was most Siberians' only immediate interaction with the Russian state, also had wide-reaching effects. The Russians coaxed yasak out of them by hostage-taking and raids, so it was very hard to say no, and some died because they could no longer feed themselves. More likely, though, they would show up to pay yasak, but do so under false names to make it harder to take a proper count and possibly demand more. So even as Siberian communities were very immediately affected by disease and military conquest, they remained surprisingly elusive and hard to pin down as subjects.
So, to get back to your questions: indigenous Siberians would certainly have been aware of the Russian presence on what had been their lands, but not because of any real administrative integration. They certainly knew that their way of life was disrupted, they were subject to raids and violence, and they were forced to pay tribute. However, in the period you're asking about, they rarely converted or came under the secular authority of the judicial system, and unless they were taken as hostages or decided to settle as peasants, they were not at all subject to Russian forms of governance outside of that tribute.
Sources:
Diment, Galya, and Yuri Slezkine. Between Heaven and Hell: The Myth of Siberia in Russian Culture. New York: St. Martin's Press, 1993.
Hartley, Janet. Siberia: A History of the People. New Haven: Yale University Press, 2014.
Slezkine, Yuri. Arctic Mirrors: Russia and the Small Peoples of the North. Ithaca: Cornell University Press, 1994.