Since you can see Alaska from Siberia, and the Siberians knew that there was apparently something over to the east, as well as the fact that they had technology like ships and telescopes. How do we know that the Siberians were not the first ones to cross the Strait and establish colonies, trading routes, etc on the Western Seaboard of America, before or during the time the Scandinavians did?
They were the first.
The Americas' indigenous population is believed to have come across the Bering land bridge (known as Beringia) during the last glaciation, which peaked about 21,000 years ago. These people advanced along the coast in boats and walked along glacier-free paths, following the herds. (They were still hunter-gatherers, remember -- no agriculture yet.)
Until recently, we believed this wide plain was simply a means of passage from one continent to the other, but a new study indicates people may have settled in Beringia until it disappeared under the rising ocean.
Now, I get the sense that you're not talking about ancient prehistory, but something within the last millennium or two.
It's important to note that until about the late 17th century, the technological level on each side of the Bering Strait was about the same. Russian influence didn't start to arrive in Siberia until the 17th century, and the first recorded passage of the Bering Strait (it's still disputed by some) didn't happen until 1639.
But you don't need modern technology to establish trading routes, and there is evidence of extensive cross-Strait trading. Archaeologists have found artifacts definitively connecting the two sides of the Strait, and linguistic and genetic evidence (as well as oral tradition) further backs this up.
A cast bronze buckle (there was no cast bronze in Alaska) was discovered three years ago near Kotzebue and is at least as old as 600 AD. That buckle could've come from China or points farther away -- we know there was no metal casting in Siberia until it was brought there in the 17th century.
What about colonization? Well, this is before written records, so it's tough to tell. We do know that the population living in Siberia and Alaska at this time wasn't the colonizing type. To create a colonizing empire, you need population pressure, among other things. (Yes, I know there are a lot of other factors.)
That population pressure didn't exist, the technological level was about the same on both sides of the Strait (it helps to have a technological disparity when colonizing an already-populated place), and there was no real reason to colonize. There was plenty of land on both sides of the strait, no resource conflicts (remember, no metal working) and existing trade routes made for friendly contact on both sides of the Strait.
There might have been occasional bouts of warfare, but I'm not familiar with any evidence that indicates mass colonization within the past few thousand years. (Until, of course, Europeans arrived.)