HikeArizona.COM

question wrote:How do I know which backpacking trails allow overnight camping?

to the best of my knowledge, enhance if you know better

Depends on the area. When unsure, call or google the managing agency such as
National Park
National Forest
Bureau of Land Management

National Parks like Grand Canyon and Saguaro require advanced planning as do some high use areas such as Fossil Creek.
Typically you can camp anywhere in a forest not in a National Park. High use areas have special regulations. Seems one general rule is to camp away from water sources.

Description pages here on HAZ have a permit/fee section.
examples
https://hikearizona.com/decoder.php?ZTN=16 has various links to regulations / permits
https://hikearizona.com/decoder.php?ZTN=116 has a map that shows restriction boundary

Suggested Reading
Intro to Arizona Backpacking / Options

Joe covered it pretty well.

On National Forest and BLM land, you may generally camp anywhere except as otherwise posted.

The restricted areas are generally those that are very popular and often near lakes or creeks.

Coconino NF has a nice page with basic guidelines for dispersed camping. Though the rules may vary somewhat from one forest to another, the others generally follow these same guidelines:

https://www.fs.usda.gov/detail/coconino ... rdb5313448

Common sense applies in many cases. If you wish to set up camp in a popular area, there may very well be restrictions of some kind. The farther you are from other people, lakes, streams, campgrounds, roads, private property, etc. the greater the likelihood you will have no problems settling down for the night anywhere you want.

State Trust Lands require a recreation permit.