This depends on their position in society, the type of event, and the distance between the regions and other factors, however in general "not well" probably sums it up. Remember in the medieval period, while news of events would travel, in general it would be quite a select audience that would hear of them, at least at first, and subsequently it would be their interpretations or understandings of the event which disseminated down to other people.
I think the best way to describe this would be an actual example.
In the 1140s, a large Mongol expeditionary force began moving westwards and came into conflict with the eastern borders of the Seljuk Empire, this culminated in a battle in 1141, the Battle of Qatwan.^(1) News of this battle reach the Latin Christian kingdoms of the Crusader States, and was carried further west into Europe. Here, the battle was believed to have been by Christians from the east, perhaps India, pushing westward to meet the "expanding" western Christians. This Eastern Christian King, named Prester John, was supposed to be head of a vast Christian realm. The Prester John legend was built and informed from what Christians of the time understood of the east, which was based on historical tales of St. Thomas the Apostles proselytising in India, as well as a western interpretation of Nestorian Christianity in Muslim-held lands.^(2)
Thus, this tale of an event, carried by churchmen, recorded by other churchmen, was interpreted and understood through their own education, and thus wider western society came to learn of the legend of Prester John, and hopes for the meeting between eastern and western Christianity to defeat the resurgent Islamic forces continued. Over 70 years later, following the disaster of the Fifth Crusade, the news of further Mongol attacks against the Khwarazmian Empire brought back by Jacques de Vitry, Bishop of Acre were interpreted by the west to be the work of King David, grandson of King Prester.^(3)
So people in the medieval period could be aware of events happening in other world regions, even quite distant, even thousands of miles away, but it was a limited understanding, heavily framed by their own understanding of the world, and interpreted through that lens.