The development of quantum networks will be paramount towards practical and secure telecommunications. These networks will need to sign and distribute information between many parties with information-theoretic security, requiring both quantum digital signatures (QDS) and quantum key distribution (QKD). Here, we introduce and experimentally realise a quantum network architecture, where the nodes are fully connected using a minimum amount of physical links. The central node of the network can act either as a totally untrusted relay, connecting the end users via the recently-introduced measurement-device-independent (MDI)-QKD, or as a trusted recipient directly communicating with the end users via QKD. Using this network, we perform a proof-of-principle demonstration of QDS mediated by MDI-QKD. For that, we devised an efficient protocol to distil multiple signatures from the same block of data, thus reducing the statistical fluctuations in the sample and greatly enhancing the final QDS rate in the finite-size scenario.