We consider how a set of collaborating agents can distributedly infer some of the properties of the communication network that they form. We specifically focus on estimating quantities that can characterize the performance of other distributed algorithms, namely the eccentricities of the nodes, and the radius and diameter of the network. We propose a strategy that can be implemented in any network, even under anonymity constraints, and has the desirable properties of being fully distributed, parallel and scalable. We analytically characterize the statistics of the estimation error, and highlight how the performance of the algorithm depends on a parameter tuning the communication complexity.