A fresh wave of state legislatures are following California in weighing data privacy legislation this year, against the backdrop of a series of data scandals involving Facebook Inc., Alphabet Inc.'s Google and other companies.

Bills of varying scope are pending in at least eight states. If enacted, the prospective laws could give companies new regulatory headaches, and up pressure on Congress to enact a federal law pre-empting the states.

“Given the nation-wide momentum around improving privacy protections, state privacy laws...