False rumors of random federal immigration raids have sent panic through immigrant communities around the country this week, emptying classrooms, work sites and shopping areas and sending thousands of people into hiding. Towns in New Jersey and on Long Island have been among those most affected by the snowballing fear, say immigrant advocates and public officials, who described terrified mothers pulling their children from a Head Start program in Freehold, N.J., and deserted streets in usually bustling areas of West Hempstead, N.Y.