The Inuit ranged from Alaska to Greenland. The Yupik were in Alaska and Russia. Would peoples have kept in contact across islands in the Arctic but not across the Bering Strait? This question could be expanded to include whether any humans have lost contact from where they came from ever, and how much certainty there could be about that one way or another (like I have read contact to and from Australia was maintained by one island to the next, I don't have a good online reference for this though). But the Yupik are one of the best examples of a people extending across a region that seems to be considered out of contact (by many, at least during pre-columbian times), with the possibility even that Yupik in Siberia back-migrated from Alaska. One could argue that the migrations were the only time any sort of contact occurred though (and that it was one way), and aside from those occasions the peoples remained unknown to each other, or only in stories. So was there possibly contact maintained across the Bering Strait, at least for certain peoples at times (precolumbian)?
On a cultural level, yes. The Yupik peoples have inhabited both sides of Bering Straight for at least a couple of millennia, though there are distinctions between the Siberian and various Alaskan groups.
Archaeological evidence on St. Lawrence Island, amidst the Bering Strait but slightly closer to Siberia than Alaska, demonstrates the cultural affinity to the Yupik peoples on both sides of the strait.
It has been inhabited intermittently for the past 2,000 years by Yupik Eskimos. The cultures of the island's population show links with groups on both sides of the Bering Strait. Extensive archaeological studies have been conducted on the island. These studies note both the archaeological and historical roles in the development of Arctic cultures.