Really the only cross-Bering people are the Inupiat Eskimos of the Diomede Islands, about 2½ miles apart. The Russia–U.S. border runs between them. In winter they can walk over to visit each other.
Chukchi are neighbors of the Siberian Eskimos, but they have no contacts on the American side. Still less Evenks, Sakhas, and Yukaghirs, who live hundreds of miles from the Bering Strait. The Ket people of the middle Yenisei are even farther away, but Ket is connected with the Na-Dene languages of America, so if any Siberians besides Eskimos had a motivation to check out their American cousins, it would be Ket people.
Some Aleuts have settled on the Komandorsky Islands, which are the other half of the volcanic island arc that is the Aleutian Islands. So Inupiat and Aleuts are the only peoples on both sides.