One of the characters of the fourth season of "Stranger Things" is a pilot flying a small plane between the Soviet Union and Alaska on a regular basis, having a business office at a US airfield. A few questions about this:
I double checked and it looks like Season 4 is supposed to take place in March 1986.
Just to work backwards: even at this point (and basically until the very last years of the USSR, no earlier than 1989 - Soviet era rules restricting travel only really started to be replaced wholesale in Russia starting in 1993) travel of Soviet citizens abroad would be heavily restricted and regulated - anyone wanting to travel would be heavily vetted by the KGB and party authoities, and would be traveling in an officially-sanctioned group (not as individuals). Friendly socialist countries would be preferred, and if you were reliable enough to qualify for travel to Western countries, you'd still probably spend a good amount of time with local Communist counterparts.
For more about travel of Soviet citizens, you might want to check out this answer I wrote in a thread with further info from u/AyeBraine and u/Dicranurus.
US travel to the USSR was possible, and it had less restrictions on the American end compared to Soviet citizens traveling abroad. But travel within the USSR was all through the purview of the Soviet agency Intourist, and was usually in groups to selected cities and/or with "minders". Think how travel to North Korea would be today. u/DrMalcolmCraig has some info in an answer here.
Lastly, as for travel between the USSR and US over the Bering Strait - this basically did not happen. The border (sometimes called the "Ice Curtain") was pretty heavily militarized, and like all sensitive Soviet borders actively patrolled and monitored. Despite the two Diomede Islands being a couple miles apart, with families having members on both islands, contact between the two was closed, and a Soviet military base established on Big Diomede.
The first official crossing between the two islands that I'm aware of was actually a 2 hour 6 minute swim (in 43 degree F / 3.3 degree C water) by long distance swimmer Lynne Cox on August 7, 1987. She had petitioned Soviet officials for years, and only was allowed to because of Gorbachev personally granting permission a mere 24 hours before her scheduled swim. Even then there were US and Soviet fighters patrolling their respective airspace just before the swim.
Speaking of 1987 and small plane travel, to give an idea of how intensely Soviet airspace was patrolled, there was a national/international incident when a then-18 years old Mathias Rust flew a Cessna from Helsinki into the Soviet Union, flying low to avoid air defenses. He managed to land in Red Square and was promptly arrested (he was imprisoned for 14 months), the incident allowed Gorbachev to fire numerous senior military officers over the international embarrassment the incident caused (the USSR was more than willing to track and shoot down any planes that strayed into Soviet airspace and did not respond to air defense demands, most notoriously the Korean Airlines flight KAL-007 in 1983).
So: no - simply put, no one would be traveling between Chukotka and Alaska in March 1986, certainly not on a routine basis. The border was heavily militarized and patrolled, US citizens would be expected to visit the USSR through official Intourist channels only, and Soviet citizens' travel abroad was heavily regulated. Anyone independently traveling over the Bering Strait would need highest level permission to do so (and basically make history, like Lynne Cox did), or would be doing so illegally and face the threat of being treated as a smuggler or spy.