Local Law Enforcement Personnel Act of 1986 - Directs the Attorney General to make grants to five eligible local governments to employ additional police officers to enforce laws against illegal drug traffic. Delineates the eligibility criteria for such grants, and for the grantee selection method. Directs the Commissioner of Customs to publish an annual report on the volume of narcotic drugs entering the United States illegally. Directs the Attorney General to solicit applications for such grants within 180 days after the date of enactment of this Act. Authorizes appropriations for FY 1987 through 1989.