Arrest information, including mug shots, is not necessarily public record. However, most states keep an online database of people in their penal system that includes a mug shot and basic identifying information available as free public records.
Not everyone who is arrested has a mug shot picture taken of them; many people are simply processed and released from police custody. Also, in incidences where the mug shot is part of an ongoing investigation it is most likely to be exempt from any public records request. A simple Web search should return whether or not a specific state provides the free online record of their penal system's mug shots.