The background check is a federal thing with federal laws regulating it. The states are supposed to report things to the feds for things to show up on the NICS. There have been failures related to the NICS due to states or military not reporting things to the feds but they don't really report stuff to other states for the background check. And if states are asked about things it's the feds doing the asking. The FBI manages the background checks and NICS.
There is also a big difference between states sharing info about convictions and a random out of state police department, ag, or governor asking a hospital to violate HIPPA for something that isn't a crime in that state.