I'm referring to the period after the land bridge disappeared, of course.
You may want to check the FAQ over there in the right-hand panel or search for previous answers. This is one of the most common Alaska-related questions on the subreddit.
There was extensive contact between people in today's Russia and people in today's Alaska. Heck, under the strictest interpretation of that border, there were (and are today) families split by the border between Little Diomede and Big Diomede islands.
Similar cultures are found between Native groups on each side of the border, and even Native languages are the same in some cases.
There's also evidence of trading between Native Alaskans and Asia. In 2011, researchers announced they had found a 1,000-year-old cast bronze buckle at an archaeological site north of the Arctic Circle. Whether that buckle came to Alaska as part of an organized trade route or simply from hand-to-hand is unknown, but bronze casting was unknown to Alaska until the Russians (and Spanish, and British, and French) arrived in the 1700s.