Surrogate-assisted evolutionary algorithms (EAs) have been proposed in recent years to solve data-driven optimization problems. Most existing surrogate-assisted EAs are for centralized optimization and do not take into account the challenges brought by the distribution of data at the edge of networks in the era of the Internet of Things. To this end, we propose edge-cloud co-EAs (ECCoEAs) to solve distributed data-driven optimization problems, where data are collected by edge servers. Specifically, we first propose a distributed framework of ECCoEAs, which consists of a communication mechanism, edge model management, and cloud model management. This communication mechanism is to avoid deadlock during the collaboration of edge servers and the cloud server. In edge model management, the edge models are trained based on local historical data and data composed of new solutions generated by co-evolutionary and their real evaluation values. In cloud model management, the black-box prediction functions received from edge models are used to find promising solutions to guide the edge model management. Moreover, two ECCoEAs are implemented, which proves the generality of the framework. To verify the performance of algorithms for distributed data-driven optimization problems, we design a novel benchmark test suite. The performance on the benchmarks and practical distributed clustering problems shows the effectiveness of ECCoEAs.