Thanks for the photos. Either badge may be a brass or copper alloy or may get its yellow color in another way such as surface plating or tinted lacquers.
The Chief of Police badge looks as though the spotty corrosion is black (at least on my monitor). The parts that are still bright appear to be protected with a lacquer or some other kind of coating but the spots may be in locations where the coating has been damaged or has otherwise failed. The black color suggests a sulfide corrosion product. If all this supposition is accurate, an objects conservator can remove the corrosion and touch in the coating to stabilize the badge. Note though that the raw metal that will be found under the corrosion is likely to have been disturbed and may have a pitted texture or an off-color after cleaning.
The star shaped badge is harder to figure out. It also looks like it has/had a coating, but this coating layer looks like it has been heat damaged or burned. Not sure what I can add here except that both are damaged, both are slowly getting worse, and there is no easy fix other than conservation. Keep them dry and cool until you can have them looked at.