Rutherford County, Tenn., faces a major public health problem. So like more than 400 other U.S. counties and cities facing health-care, emergency-response, and other costs tied to the opioid crisis, Rutherford is taking matters into its own hands. With some help, that is, from some seriously big hired guns.

The county recently filed suit against Purdue Pharma L.P., Johnson & Johnson, Janssen Pharmaceuticals Inc., Insys Therapeutics Inc., and a spate of other drug makers and distributors alleging public nuisance and...