This is one of those questions that, while incredibly interesting, is hopelessly difficult to answer, due to the nature and extent of the sources. The Greek sources that exist impose a cultural filter on their understanding of the functioning of the Achaemenid court that we must attempt to see through, while we at the same time lack the understanding of Achaemenid society that we would need to really do so.

The most useful Persian source on the matter is the collection known as the Persepolis Fortification Tablets, from the reign of Darius I. These are brief communications dealing with the logistics of the royal estates as well as the coordination of substantial (probably conscripted) labour forces. They typically take the form of a communication from a high-ranking member of the royal family (e.g. Parnaka, the uncle of Darius) to an administrative official (e.g. Irtuppiya) to commision goods (e.g. wine, grain, livestock) for a recipient (an estate, a labour force, etc). There are interesting facts that can be teased out of these documents (such as the fact that women seem to have had a near-equal status in the labour force, based on compensation and the fact that they could head substantial groups of labourers), but they are very limited, being only one step in an administrative ladder.

They do reinforce the notion, seen in other sources concerning e.g. marrying into the royal family, that authority generally stemmed from familial (or at least social) relations with the Great King. But they are not just limited in vertical extent - they are also centered on the heartlands of the empire. Clearly, by the massive construction works undertaken, Darius and his successors must have been able to count on regular and substantial tribute from their subject territories. The general sense one gets is that the organization of the realms into satrapies was not so much a matter of exerting direct royal control over territories (as in a modern absolutist monarchy) but instead a matter of levying tribute effectively. Herodotus somewhat notoriously gives a breakdown of the tribute levied from the various territories, although we should probably be careful not to attach too much value to his assessment. Though, Pierre Briant notes in "From Cyrus to Alexander", Herodotus' numbers here do appear to be in the right ballpark, to the extent that Briant suggests Herodotous had access to administrative records of some sort. In addition to gold, it is asserted that some terrritories were allowed to pay their tribute in the form of luxury goods like ivory or incense, or in the form of young men and women.

We can probably be safe in assuming, as Herodotus suggests, that careful accounting was an important facet of Darius' administrative reforms and a key reason for the effective administration of the empire. Herodotus suggests that Persia itself was not taxed - this is obviously false based on assessments in Persepolis tablets, but we shouldn't be too alarmed at this contradiction, which is probably the result of an ideological-political distinction. It does seem plausible though that the royal household managed what one might call its demesne in a different, more micromanaged manner, leaving the satrapies to their own devices in figuring out how to pay the predetermined tribute. Pseudo-Aristotle's Oeconomicus indicates that in addition to fixed tribute, various taxes on production and commerce were levied as well - taxes on artisans, cattle, agriculture, mining, et cetera. As suggested earlier, peasants could also be pressed into corveé, that is, conscripted as labourers on behalf of the state; this is best attested in Babylonia.

It appears, finally, that a particularly extravagant notion of hospitality was an important part of the king's clout. Whenever the king was moving his army through a territory for a military expedition or attending to some other project, it seems to have been customary for the entire resources of the satrapy to be placed at his disposal as a form of "hospitality", and for wealthy nobles to present the king with enormous "gifts" of wealth - which, obviously, they would expect to be compensated for in the form of royal favour.

Sorry if this is a bit hasty, but I need to leave for work, so I don't have time at this very moment to start combing through primary sources for specific quotations and the like to give you. I'm happy to take any followup questions and elaborate on any points later. But, as you can see, the available evidence is quite limited.

This answer was primarily based on chapter 10 of Briant's "From Cyrus to Alexander" and familiarity with most of the primary sources used.