First, crows are a protected species under the North America Migratory Bird Act. Any type of lethal control requires appropriate permits available from state and federal agencies.
Shooting is more effective as a dispersal technique than as a way to reduce crow numbers. Crows are wary and thus difficult to shoot during daylight hours. They may be attracted to a concealed shooter, however, by using crow decoys or calls or by placing an owl effigy in a conspicuous location. Generally, the number of crows killed by shooting is very small in relation to the numbers involved in pest situations. However, shooting can be a helpful technique to supplement and reinforce other dispersal techniques when the goal is to frighten and disperse crows rather than specifically to reduce numbers.
Crow hunting during open season can be encouraged in areas where crows cause problems. The helpfulness of hunting as a control technique varies depending on crow movements, the season in which the damage occurs, and other factors. If using a shotgun, use #6 shot.
Another consideration is that crows tend to be more wary of people when they are hunted and thus more easily dispersed from roosting or other areas where their presence is a problem. Further study is needed to better understand the relationships between hunting and wariness, and whether a pattern exists that might be used to improve crow management programs.
Resources: (online)