In effect, it would entirely depend on the person's race, gender, social, and disability class or status.

Generally speaking, the population most likely to have experienced a formal history education would be White men with access to disposable income and/or power. By the mid-1800s, history was increasingly a topic of study by itself (up until that time, a classical education meant a young man would study texts by historical figures in the context of learning logic and rhetoric.) So, a young man from a family of means who attended one of the Colonial Colleges before heading west was likely well-educated in history.

The people least likely to be educated in European history were those farthest from men with access to power. For example, a recently free enslaved woman who made her way West likely did not have a history education as basic literacy was denied her while she was enslaved. However, if she attended a school sponsored by the Freedman’s Bureau immediately following The Civil War, she may have come across the names you mentioned as well as American history, based on the texts her teachers had available to them.

It's possible an Indigenous person, especially a boy or man, might be familiar with historical figures if they attending one of many Indian Boarding schools that were established as a way to indoctrinate Indigenous children and break their language and cultural bonds to their communities. Children were often expected to memorize historical texts and in some cases, were given access to libraries that included books on world history.

Meanwhile, a young White woman would probably have some exposure to a history education, depending on what brought her across the country. If, for example, she was there as a newly hired teacher, fresh from an East Coast teacher prep school, she may very well have taken a class on how to teach history in her school and learned world history in the process. Or, if she came from wealth, she may have had a private tutor who included history in her studies.

All of that said, it is entirely possible a random person living on the American frontier may have known who those men were because they had a history textbook in their personal library or read a fictionalized take about their lives. They wouldn't have, though, studied history in the way we think of it in the modern era. In this previous answer, I go into the history of history class and how the notion of studying history became part of the modern liberal arts curriculum.