Is there any public land where they do Mars Rover testing?
Yes, there are public and publicly accessible lands used for Mars rover testing.
The Mojave Desert in California is a primary location where NASA's Jet Propulsion Laboratory (JPL) conducts rover field tests. Sites like Dumont Dunes, near Death Valley, have been used to test full-scale mockups of rovers like Curiosity (using a rover called ”Scarecrow”) to practice driving on sand dunes and other Mars-like terrain.
Another key site is the Mars Desert Research Station (MDRS) near Hanksville, Utah. This area, managed by The Mars Society, is a designated Mars analog environment. It features a barren, red-rock desert landscape that closely resembles the surface of Mars and is used for testing rover operations, conducting scientific research, and training crews for simulated Mars missions. The annual University Rover Challenge (URC) is held here, where student teams test their own rovers.