Originally proposed for privacy protection in the context of statistical databases, differential privacy is now widely adopted in various models of computation. In this paper we investigate techniques for proving differential privacy in the context of concurrent systems. Our motivation stems from the work of Tschantz et al., who proposed a verification method based on proving the existence of a stratified family between states, that can track the privacy leakage, ensuring that it does not exceed a given leakage budget. We improve this technique by investigating a state property which is more permissive and still implies differential privacy. We consider two pseudometrics on probabilistic automata: The first one is essentially a reformulation of the notion proposed by Tschantz et al. The second one is a more liberal variant, relaxing the relation between them by integrating the notion of amortisation, which results into a more parsimonious use of the privacy budget. We show that the metrical closeness of automata guarantees the preservation of differential privacy, which makes the two metrics suitable for verification. Moreover we show that process combinators are non-expansive in this pseudometric framework. We apply the pseudometric framework to reason about the degree of differential privacy of protocols by the example of the Dining Cryptographers Protocol with biased coins.