Sauron
It's not entirely clear. It does seem that, at least initially, the Haradrim and Easterlings were directly under Sauron's control, but did have their own forms of government (emphasis mine):
Sauron gathered to him great strength of his servants out of the east and the south; and among them were not a few of the high race of Númenor. For in the days of the sojourn of Sauron in that land the hearts of well nigh all its people had been turned towards darkness. Therefore many of those who sailed east in that time and made fortresses and dwellings upon the coasts were already bent to his will, and they served him still gladly in Middle-earth. But because of the power of Gil-galad these renegades, lords both mighty and evil, for the most part took up their abodes in the southlands far away; yet two there were, Herumor and Fuinur, who rose to power among the Haradrim, a great and cruel people that dwelt in the wide lands south of Mordor beyond the mouths of Anduin.
The Silmarillion V Of the Rings of Power and the Third Age
Recall that Sauron's great strength is the ability to impose his will upon others; whether or not he's personally involved in their governance, the kings who were in charge are acting in accordance with his wishes at all times.
But that's the Second Age, which features Sauron's defeat and his thousand-year banishment. In the Third Age, it's possible he's taking a more hands-off approach (emphasis mine):
The Nameless Enemy has arisen again. Smoke rises once more from Orodruin that we call Mount Doom. The power of the Black Land grows and we are hard beset. When the Enemy returned our folk were driven from Ithilien, our fair domain east of the River, though we kept a foothold there and strength of arms. But this very year, in the days of June, sudden war came upon us out of Mordor, and we were swept away. We were outnumbered, for Mordor has allied itself with the Easterlings and the cruel Haradrim; but it was not by numbers that we were defeated. A power was there that we have not felt before.
Fellowship of the Ring Book II Chapter 2: "The Council of Elrond"
This is Boromir speaking, and we must acknowledge that his information isn't perfect; it seems unlikely that he knows much about the finer nuances of Mordor's political manoeuvres. However, his use of the word "allied" implies a more feudal situation than in the Second Age; the lords of the Haradrim and Easterlings swearing loyalty to Sauron and his cause, but otherwise self-govern. Without more information about the Mordor political machine, which we don't have, we can't say a lot conclusively.
However, we do know that at least Harad was self-governing in Sauron's absence; to the point where they had peace with Gondor (something Sauron would never have approved):
'Tis said that there were dealings of old between Gondor and the kingdoms of the Harad in the Far South; though there was never friendship. In those days our bounds were away south beyond the mouths of Anduin, and Umbar, the nearest of their realms, acknowledged our sway.
The Two Towers Book VI Chapter 4: "Of Herbs and Stewed Rabbit"
Morgoth
There's evidence for both; it's said that Morgoth sent messengers among early men, but he wasn't above using torture when peaceful overtures failed:
To Men Morgoth feigned pity, if any would hearken to his messages, saying that their woes came only of their servitude to the rebel Noldor, but at the hands of the rightful Lord of Middle-earth they would get honour and a just reward of valour, if they would leave rebellion. But few men of the Three Houses of the Edain would give ear to him, not even were they brought to the torment of Angband. Therefore Morgoth pursued them with hatred; and he sent his messengers over the mountains.
The Silmarillion III Quenta Silmarillion Chapter 18: "Of the Ruin of Beleriand and the Fall of Fingolfin"
But it also appears as though at least some (though not all) of the Easterlings were under Morgoth's direct control:
It is told that at this time the Swarthy Men came first into Beleriand. Some were already secretly under the dominion of Morgoth, and came at his call; but not all, for the rumour of Beleriand, of its lands and waters, of its wars and riches, went now far and wide, and the wandering feet of Men were ever set westward in those days.
The Silmarillion III Quenta Silmarillion Chapter 18: "Of the Ruin of Beleriand and the Fall of Fingolfin"
So it's entirely possible that Morgoth was doing a little of both, at least at first, but I suspect he would ultimately swing towards total domination; Morgoth isn't very good about sharing his toys.
