Can't tell you much specifically about Ohio, but in AZ the answer would depend on the type of public land. For State Trust land target shooting is strictly forbidden, however hunting is allowed. For BLM and National Forest land target shooting is generally allowed, but especially in the case of National Forests the NFS can restrict target shooting to discrete portions. That is mostly for the safety of others since unfortunately there are some idiots that'll blast away in unsafe areas otherwise. State Parks are usually not available to target shooters (though CCW rules vary, in AZ having a permit allows you to carry in a state park - municipal parks are not as clear). National Parks are, of course, off limits to guns whether or not they're being discharged for any reason (including self defense) almost entirely (there are some limited circumstances mostly related to "just passing through with an unloaded, locked gun", law enforcement, and game herd culling efforts where that statement doesn't apply).
