At what point in time was such long distance travel thought necessary, and how did they learn each other's languages? Would the average person have known about the discovery and what was going on at the time, or did they always have some sort of awareness of the other one?
Was there a point where contact was cut off, and what happened to lead to a rediscovery?
/u/cronos844
Are you asking about when China and Western Europe met in person, or "met" as in, had some degree of contact.
The trade of valuables between Europe and China is very old. Or to be more precise, silk fragments have been found in Egypt dating back to around 1000 BC and believed to have come from China. And prior to this time, gems from the Middle East have found their way to China. With the Roman conquest of Egypt, a link was established from China to the Red Sea, and then onto the Mediterranean and the rest of Europe. In addition to the land route that benefited through investments by the Tang dynasty.
The Roman link is important, because both China and the Roman empire were aware of each other's existence even if details were veeeery fuzzy. Both sides attempted to send envoys to each other, around the 1st-2nd century AD. The Roman empire at this time had dominion over parts of Western Europe. By this time, Chinese silk had found its way to the Roman Empire.
Want to know more? Wood's The Silk Road: Two Thousand Years in the Heart of Asia, Liu's The Silk Road in World History.