I've also heard that people in the Arctic circle were aware that there were others across the Arctic ocean but I'm not familiar if there was any contact after the Bering Sea severed the land bridge.

Trans-Bering contact was essentially permanent in the region. The Yupik peoples, in particular, live on both sides of the Bering Strait, and have a long history of interaction with each other and their neighbors in both North America and Asia.

As for Pre-columbian expeditions eastward across the Atlantic, we don't have notable evidence of that. While there were strong maritime traditions in the vicinity of the Gulf of St. Lawrence and the Caribbean, neither area had boats well suited for a long-haul voyage across the Atlantic.

Post-Columbus though, we do have some examples of explorations sponsored by Native communities to Europe, taking advantage of the European ships sailing back and forth already.

A notable example would be Tomocomo's expedition to England in 1616-17. He had been sent by Wahunsenacawh (aka "Chief Powhatan") to count and observe the English in their homeland. This was the same voyage that carried Pocahontas to England.

The Powhatan conception of England at the time of colonization was that it was a small, impoverished island whose people were in desperate need of assistance. It did not take long for Tomocome dispel that misconception and grow tired of counting every English person he saw. While he never had a formal audience with King James, Tomocomo did bump into him at a performance one night and was later upset that no one told him that that was king. Overall he found the hospitality of the English lacking and realized that they did not regard the Powhatans as allies in the same way that the Powhatans regarded them. Upon returning to Tsenacommacah (Virginia), he mounted vocal opposition to the English presence there.

My degree specifically focused on pre-columbian and early colonial Central America and Mexico. There are no stories I am aware of but there is some references to other peoples in other far away lands. It's not really much to go on, however. In the Americas there is evidence that linked groups knew of each other and even had some rough ideas of where tribes far to the north or south were located. That's pretty generally though. It's also how parrot feathers end up in head dresses as far north at the Dakotas.

Specifically, the real issue is the lack of material evidence for large ships. Even a trip across Brasil to Africa requires a large ships capable of withstanding some intense weather. If these groups had ships like the Pacific Islanders then it could be totally possible as Pacific Islanders traveled hundreds of miles on catamarans that could be quite large. However, no ships have been found that could survive over seas exploration. The Maya, for example, had the right knowledge of stars but again no ships.

A final note would be asking what these people expected to get out of a big boat trip. They lacked nothing and lived in a resource rich land. It's easier to see why Europeans needed to go out taking land and making trade than say Africa or the Americas which had everything one would need save for large animals for farming (in America that is unless you want to breed a super llama).