It's difficult to talk in generalities about people from an entire continent over a span of 500 years.

Some people very much did. For a start, they weren't always never seen again. There was migration back from the New World by ex slaves. This is mostly something you see from 1790 to 1888 during the last gasp of it. Sierra Leone and Liberia were settled by New World slaves and had regular contact with other African polities. And they weren't even all that separate from the existing power structures. The Afro-Brazilians who were deported back to Africa in the 1830s after a slave revolt was put down mostly moved into and settled in existing African Kingdoms. If a slave trader in 1840s Benin wanted to know what happened to the slaves he sold, he could ask the ex Brazilian slave in the hut down the road.

Even prior to this, it was not rare for slaves and ex slaves to be represented on board ships trading in Africa. Abraham Samuel was born in a slave colony in the New World and travelled to Africa on board an American pirate ship in 1690s. He would later become King of an area of Madagascar, so again if any of the Malagasy in that region were interested in to what happened to the slaves they sold, they could ask him.

They also sometimes made visits to the New World themselves. When in the 1740s the son of a African Slave Trader was double crossed and himself sold into slavery, he was rescued from the plantation by a companion of his being toured around the West Indies plantations by the British Navy until he recognised the son who could be freed (the son would still act as a slave trader upon his return to Africa). So within that time and place, 1740s Ghana, you had among the elite slave traders, at least one who had visited plantations personally and at least one who had been a plantation slave. So again if someone in Ghana in that region wanted to know, they had people they could ask.

And the way in which the British allowed that to happen, brings us to the way the elite Africans normally were quite seriously wooed by the Europeans because a boycott was disastrous. During the time the son was enslaved, his father, John Corrente, started trading exclusively with France and encouraging women with incurable stds to seduce British sailors while in port. They had good economic reasons to get him back on side by getting the son back from the West Indies (and indeed hosting the son in England for several years as an apology, with him being a guest of high society).

The elites were sometimes given European education, they sometimes had personal correspondence with European Monarchs about the slave trade (see the letters between the Kings of Portugal and Kongo about reducing the slave trade in 1526) and again for those elite there'd be no question that they knew what was happening. The traders were often operating within native Kingdoms and were at the mercy of those leaders, there's little to no evidence of demands to be informed about where those people are going not being unanswered. There's no real reason for the Europeans to lie if asked.

Would everyone know? No. There's plenty of evidence of folk tales about the White People eating the slaves etc. But the more involved you were within the trade, the higher your status and the later the year is, the more likely it is that you would know. Especially once you get into the 19th century and the first abolitionists and African American freemen start visiting African leaders, knowledge can be assumed for anyone in a position of power.

u/youngmarshall's answer is excellent, so I'll only add a few notes around the edges of it.

In early Atlantic exchanges between the Portuguese and societies in Senegambia, Upper Guinea, Kongo and Angola, African rulers often had tangible knowledge of what life was like in Portugal and Western Europe, and at least some of the relatively small numbers of enslaved persons caught up in those early trades went to the Atlantic islands rather than the New World, which made answering the question of where the slaves were going more tangible and grounded. As Toby Green observes in his recent work, in the earliest development of Atlantic trade, West and Central African societies substantially assumed that slaves were being incorporated into Portuguese society (or Portugal's overseas possessions) in ways that were not too dramatically different from the social status of slaves in their own societies. This wasn't a correct assumption, but it meant that it wasn't treated as mysterious.

As the scale and intensity of the Atlantic slave trade climbed at the end of the 17th Century and into the 18th Century, this became a much more pressing question throughout West and Central Africa. One fairly common thought that many African communities had was that Europeans were cannibals who were murdering and devouring the people they purchased, a belief that intensified the fear and despair of those taken onto the slave ships. Metaphorically, this doesn't even feel all that far off: people died in huge numbers before even making it to the Americas, their bodies and lives were used up cruelly if they arrived alive as slaves. Orlando Patterson's interpretation of Atlantic slavery as "social death" is in this sense pretty on target. This thought showed up in a variety of ways--sometimes as a deadly serious concrete accusation leveled by skeptical rulers or angry opponents of the trade, sometimes as a kind of folkloric/mythological reading of the slave trade and European motivations, sometimes as a kind of consciously ironic or mocking remark by Africans who were involved in the trade themselves.

But also in the 18th Century and early 19th Century, it became more common (in part simply because of the sheer volume and density of Atlantic connections) for individuals to return from being enslaved. Randy Sparks' The Two Princes of Calabar focuses on one especially compelling example of two men involved in the slave trade who were betrayed by rivals, sold to slavers and who eventually returned to Calabar via the UK. We don't know what happened after that (Sparks speculates that they may have returned to slave trading). Several rulers of Dahomey expressed active interest in getting more information about the slave economies of the New World, and wrote letters and sent emissaries seeking this information. Michael Gomez' Reversing Sail and James Campbell's book on the history of contacts between the African diaspora and Africa detail many other cases of people returning out of slavery. In the early 19th Century, there were also more Europeans locally resident in trading ports who provided fairly concrete information about the sum total of the Atlantic trade, so for many people living in Atlantic Africa, the fate of the people taken into slavery became an increasingly known and knowable thing.