A state can auction or license the services of providing electronic ID to private companies instead of doing on its own.
Sub-contracting to private companies might result in cost-savings and benefit to citizens from marketing competition: The state can define quality requirements and appoint one or more distributors Citizens can select between several providers on the basis of costs and services provided
The risks behind a free-market approach are several. As an example: The private companies in order to offer competitive prices might lower the quality thus comprising also privacy and exposing customer to threats. A cartel can emerge thus depriving citizens of a real choice.