The Arizona State Land Department has an interactive land viewer here:
https://land.az.gov/maps-gis
Under the layers option you can turn on "land ownership". That will tell you who owns what areas so you can check that agency web site for fire and shooting restrictions. Most NF and the BLM close their lands to shooting / fires at the beginning of the summer and rescind the restriction at some point at the end of the summer which can still go pretty late in the year.
As an example, Tonto NF to the east and north of Phoenix has a lot of land people would classify as desert. If you Google "Tonto National Forest fire restrictions" you get to this page: https://www.fs.usda.gov/alerts/tonto/alerts-notices.
Select "Tonto NF Fire Info" on the right side of the page. That takes you to this page:
https://www.fs.usda.gov/detail/tonto/home/?cid=STELPRDB5269295
Which says "Stage 1 Fire Restrictions are in effect on the Tonto National Forest."
Scroll down to see what Stage 1 means:
#3 is "Discharging a firearm, air rifle, or gas gun. 36 C.F.R. § 261.58(m)."
Other NF in Arizona have their own web pages with similar information.
You can always call the district office of the specific NF although they may or may not provide accurate information.
There are NF restrictions on the types of targets you can use.
https://www.fs.usda.gov/alerts/tonto/alerts-notices/?aid=88727
Here's the BLM page on shooting. Some good advice here about backstops, etc.
https://www.blm.gov/programs/recreation/arizona/recreational-shooting
Here's the BLM page with fire restrictions.
https://www.blm.gov/programs/public-safety-and-fire/fire/regional-info/arizona/fire-restrictions.
BLM and NF are the two main types of public land available for target shooting. National Monuments can have some available areas but the BLM is restricting those. National Parks say no. Indian Reservations are a big no. Military land, no. State Parks, no. State Trust Land, no target shooting, yes to hunting. Private land is, well, private. Municipal land, no.
Note that ranchers may have grazing rights on BLM, NF and State Trust Land and may have their own ideas about whether shooting is allowed. The rules of the agency should supersede what the rancher thinks but if it's you and them out in the desert you may want to take it elsewhere.
Firearms in National Parks.
https://www.nps.gov/articles/firearms-in-national-parks.htm
Using Google Maps satellite view, the ASDL land ownership viewer and the agency pages, identify an area to research. The further from a major metro area the higher likelihood of success.
Plan on taking a recon trip(or trips) to evaluate spots because some may be posted no-shooting but not indicated as such on an agency web site. The BLM does this.
Be aware that Arizona has laws about unlawful discharge of weapons, in particular the ones about shooting within municipal limits and distance to an occupied structure, which apply in addition to whatever rules the land agencies have.
https://www.azleg.gov/ars/13/03107.htm
If you do find a good spot, tell no one. Don't shoot plants, animals, structures, power poles, petroglyphs, windmills, irrigation equipment, lights, signs or anything else you didn't bring. Clean up after yourself.
