The citizenship requirements to be a law enforcement officer with a local or state agency will depend upon the individual agency. Some agencies require applicants to be US Citizens or US Nationals. Other agencies open up the requirements to those who are resident aliens. There are agencies which only require applicants to be lawfully authorized to work in the United States. Employment with a Federal Law Enforcement Agency does require the applicant to be a US Citizen or US National.
Citizenship requirements are usually posted on the vacancy announcement advertising the position. Specific requirements regarding citizenship can always be obtained by contacting a recruiter from that agency, or from the personnel department.
Each state sets the minimum requirements for becoming a peace officer in their state. It may be beneficial to find out the POST (Peace Officer Standards and Training) requirements regarding citizenship for the state in which you are interested in working as a police officer. Local agencies may set more restrictive standards than the state, but not less restrictive. Therefore, if you want to determine if your citizenship status may be a factor in finding work as a police officer, if you meet the state requirements you know there is the possibility of meeting local requirements.