As with so many things across an entire continental margin and 450+ years, 'it depends.' It depends on the closest political entity, it depends on the time within the Atlantic age, and it depends on where along the coast you're talking about.

For example, early on most of the coastal merchants on the African side would be Africans--possibly connected with buying captives from others further inland, or from a state if such a structure for authority ownership existed--but sometimes working independently. The castles that emerged in the late 1400s like Elmina were meeting places that lacked the ecological peril of traveling inland for Europeans, and secured for the major power privileged trade status. Sometimes other kingdoms and coastal polities made deals with other traders (see Rushforth's recent work on French in 16th/17th C. West Africa) and occasionally a state did keep very direct control (as in the kingdom of Benin, today in western Nigeria).

As the trades developed, a lot of the mediating role fell to Eurafricans and acclimated Europeans as well. The signares of Goree island (imagine the accented e--my computer doesn't like making them today) were Eurafrican women who effectively controlled trade flows through the island including the slave trade. Likewise, despite the creation of a Portuguese colony at Luanda to channel goods and captives outward in place of a less than reliable Kongolese state (and sometimes to raid it), many of the key figures were in this same mixed group of 'factors.' Almost anywhere on the African littoral where this trade happened, such variably metis communities evolved, and control over trade was significant there. They could engage in king or queenmaking (and often did in what today is Angola and the western DRC; see Thornton's Kingdom of Kongo) and they didn't always have a true stranglehold, but the hybridity of cultural connections is fascinating. George Brookes's old books are still well worth consulting on the development and activities of these liminal yet vital communities.

These Eurafrican (in heritage or in terms of cultural position) factors continued to have important and even controlling roles as late as the nineteenth century, and their activities lead to some of the impositions of colonial presence in places like Lagos (1861) and Zanzibar (1890). I'll see if I have something on the latter that is both good and newer than Ned Alpers's old Ivory and Slaves that explains the evolution of the entire east coast trades with many of these plural societies being Swahili with some Portuguese and Omani presences at various times. They could always still be Africans from further inland, or Europeans of long tenure who married locally (such as the problematic de Souza clan in Ouidah, the slaving port of the Kingdom of Dahomey - warning soft paywall), but usually the interfacing traders themselves could vary in their level of entanglement with powers of the interior, because most of the traders were themselves independent to varying degrees. Sometimes they were not independent at all; that's how the kingdom of Benin stayed out of the export slave trades for centuries until civil war broke that down for a time, and how the Akan states were able to remain almost constantly net importers of human beings except during war--and war was when most captives were taken. In any case, control over the mercantile activity within the major transshipment points was always a vital bone of contention.

Thornton's Africa and Africans in the Making of the Atlantic World 2nd ed (1998), the first few chapters, get into some of this variability although he is quite beholden to the narrative of Portuguese dominance that several scholars are now upturning in really interesting ways. The African side of the equation is also even more complex than he allowed, which is another 'it depended' which I know is annoying to hear in an answer. (I don't have a copy of Thornton's Cultural History of the Atlantic World from 2012 here at home, but he does try to be more nuanced about exchange there in his African chapters. That may be of interest to you.)

[edit: clarified one detail, fixed a spelling error, fixed a bad grammar me fail no the impossible did.]