How did the Aztecs and Mayans control the groups they dominated? I imagine sort of soldiers sent around to collect tribute from small villages, is this at all accurate?
The Aztecs (see my other comment in this post for the oddity of the term) did not send troops around to collect tribute. There were permanent garrisons maintained at certain strategic site, but, like so many other pre-modern states, they did not utilize a standing army, instead calling up soldiers as needed. The garrisons were there to provide a sort of rapid response to any uprising; incursion from outside groups; and/or join up with a conquering Aztec army passing though.
Instead, there was a specific professional class of tribute-collectors, the calpixque, who would live in the tributary cities and towns. Though typically translated as "tribute/tax collector," the term calpixque can actually be more literally translated (following Karttunen 1983) as majordomo/steward, or "Keeper of a House." In the Spanish texts as the calpixque are routinely referred to as "stewards" or "ambassadors."
We can see the complicated nature of their role with regards to the calpixque who were served as the first indication the Spanish received, during the 1518 Grijalva Expedition, of a large and powerful inland state and were the first to carry news of their arrival to Tenochtitlan. The second chapter of Sahagún's General History of the Things of New Spain, Book 12, deals with this:
And when were seen those who came to the seashore, already they were traveling by boat. Then in person went Pinotl of Cuetlaxtlan, the high steward. He took with him still other stewards... The Spaiards called out to them: they said to them: "Who are you? Whence have you come? Where is your home?"
Then [the stewards] said: "It is from Mexico that we have come."
The next couple chapters describe how the "stewards" (calpixque) traveled to Tenochtitlan to describe what they had seen and how Motecuhzoma ordered a watch to be kept for the arrival of more strangers on the shore. A similar dynamic occurred when Cortés arrived the following year. To quote from Díaz del Castillo (Carrasco 2008 trans.):
It appears that Tendile [a calpixque] brought with him some clever painters such as they had in Mexico and ordered them to make pictures true to the face and body of Cortés and all his captains, and of the soldiers, ships, sails and horses, and of Doña Marina and Aguilar, even of the two greyhounds, and the cannon and cannonballs, and all of the army we had brought with us, and he carried the pictures to his master.
The calpixque, in other words, were no mere tax collectors, but served as representatives of the power of the Aztecs. Their key role was to ensure tribute kept flowing, but they were also the eyes and ears of the Aztecs. In the later Aztec period, when class distinctions were growing stronger and more codified, this role would often be filled by the minor nobility, we acted as the administrators of the increasing widespread Aztec hegemony.
As for the more practical attributes of tribute, the best source we have on the various subdivisions of the Aztec state, and the tribute owed by each, the Codex Mendoza. It is a post-Conquest document (very few pre-Conquest documents survive, and even less from the Aztecs) made in the Aztec style with Spanish glosses. In other words, a typical page will have the name-glyphs of each city in particular region, as well as a pictographic representation of the tribute items owed, all of which will be annotated with Spanish text for each item. It's a terrific resource and Berdan and Anawalt's The Essential Codex Mendoza is a wonderful interpretation of the primary source.
It is because of the Codex Mendoza that we know that the "province" of Cuetlaxtlan (where Grijalva put ashore) was to supply annually:
A rich province, we have other textual evidence that the are around Cuetlaxtlan had also had tribute demands of gold, exotic skins, conch shells, and dried fish, in addition to demands recorded at the time of the Coedx Mendoza. The Codex also records that the province was required to send to the Aztecs twice a year almost 3000 cotton mantas (cloaks) in various styles, along with 400 women's tunics and skirts. This follows a general pattern in which more luxurious items would be demands annually and more staple good would be demanded in biannually. The calpixque were there to make sure this tribute occurred, cajoling the local rulers who would themselves benefit from this tribute system though "re-gifting" and to report on any problems which may be arising.
Calpixque had a vested interest in making sure cities were pacified and amenable to sending tribute, as they were often the first victims of an uprising. A fairly common tactic was actually to send emissaries to a region demand tribute and then, when they were killed, to use that as an excuse for retaliation. This is actually how Cuetlaxtlan was made an Aztec province to begin with. The Tlaxcalans convinced them to not pay tribute and to murder the stewards sent asking for it, then completely failed to back up the cities of the region when the Aztec army retaliated.
The threat of violence was always backing up the institutional role of the calpixque. In a famous instance from the reign of Ahuizotl (1486-1502), several towns in the Tepequacuilco province, in the border region of modern day Guerrero with Michoacán, refused to pay tribute. The texts do not record what happened to the calpixque of the area, but precedent suggests they would have been killed. In response, the Aztec army rolled in, offered the rebellious towns a chance to resume paying tribute and, when that was denied, leveled the key rebel towns of Oztoman and Alahuiztlan. All the men were killed, and the women and children were sold into slavery. Couples from the core Aztec cities were then selected to resettled the region. There's much larger socio-political causes behind this, but the message was clear: pay your taxes.
This a good question for the Aztecs, though I am not qualified to answer it. For the Maya, however, it's not entirely relevant. There was never a Maya "empire" or large centralized state in the area. Instead, the Maya political map was a mosaic of influential city states in a frequently shifting web of alliances, rivalries, and trade. Tikal, one of the first major superpowers, gained influence largely through control of strong trade routes, marrying into other royal families, and developing an enormous urban center. It didn't necessarily control these city other states, but they were dependent on Tikal and would (usually) respect its interests. Warfare did often occur between competing cities, such as Tikal and Calakmul, but our typical conception of inhabiting the new territory doesn't seem to apply. Usually the old king was disposed of and a new one sympathetic to the victors was installed, and the city existed as a separate vassal state.