We show that tensor networks, widely used for providing efficient representations of quantum many-body systems and which have recently been proposed as machine learning architectures, have especially prospective properties for privacy-preserving machine learning. First, we describe a new privacy vulnerability in feedforward neural networks, illustrating it in synthetic and real-world datasets. Then, we develop well-defined conditions to guarantee robustness to such vulnerability, and we rigorously prove that these conditions are satisfied by tensor networks. We supplement the analytical findings with practical examples where matrix product states are trained on datasets of medical records, showing large reductions on the probability of an attacker extracting information about the training dataset from the model's parameters when compared to feedforward neural networks.