It varies depending on what region we're talking about. The most famous communication network in the Americas was the messenger system of the Inca. Utilizing the immense road network within the empire, messengers would travel on foot. They'd stop at specially maintained inns along the way to rest and sometimes pass the message off. It's reported that news took at most a week to reach the capital from the farthest corners of the empire, a time not easily beaten until rails were constructed hundreds of years later.

Far to the North in what is now the American Southwest, the terrain is quite different. Although roads and trails were extensively used for messengers, the terrain of the area is characterized by isolated mountains and broad intermontane regions that are ideal for smoke/fire signals.

The Aztecs would use all of these as well as merchants to gather information, but they also had written and symbolic records that could be used to communicate specific types of information. Censes, tax obligations, and more would all be communicated on paper.

Every group had a secondary means of communication via trade. Trade was conducted on foot and relatively slow, but goods [and potentially messages] could cross thousands of miles this way. For the Aztecs in particular, merchants were a primary and useful way to gather information on allies and enemies.

The most exciting method I've ever heard of and one that there is exactly zero evidence for in the precolumbian Americas was the use of signal mirrors. Polished pieces of stone or metal would be used to flash messages between mountaintops in the North of Mexico, rapidly transmitting information hundreds of miles in minutes. A specially trained group of signallers who closely guarded their language were responsible for this. This reportedly occurred in the Ponca and several other groups. I'm not going to cite sources here because frankly there aren't any reputable ones for this claim in the prehispanic era. It's simply fun to talk about.

Jabusch, D. M. (1992). Communication Media in Ancient Cultures.

Beers, W. (2012). All Along the Watchtower: Prehistoric Signaling Behavior in the Jumanos Pueblo Cluster, Torrance County, New Mexico (Doctoral dissertation, Eastern New Mexico University).