Diffusion adaptation is an efficient strategy to perform distributed estimation over networks with streaming data. Existing diffusion-based estimation algorithms require the knowledge of analytical forms of the cost functions or their gradients associated with agents. This setting can be restrictive for practical applications where gradient calculation is difficult or systems operate in a black-box manner. Motivated by the advance of the zeroth-order (gradient-free) optimization, in this work we propose the zeroth-order (ZO) diffusion strategy using randomized gradient estimates. We also examine the stability conditions of the proposed ZO-diffusion strategy. Simulations are performed to examine properties of the algorithm and to compare it with its non-cooperative and stochastic gradient counterparts.