Artificial intelligence (AI) models are increasingly used in the medical domain. However, as medical data is highly sensitive, special precautions to ensure the protection of said data are required. The gold standard for privacy preservation is the introduction of differential privacy (DP) to model training. However, prior work has shown that DP has negative implications on model accuracy and fairness. Therefore, the purpose of this study is to demonstrate that the privacy-preserving training of AI models for chest radiograph diagnosis is possible with high accuracy and fairness compared to non-private training. N=193,311 high quality clinical chest radiographs were retrospectively collected and manually labeled by experienced radiologists, who assigned one or more of the following diagnoses: cardiomegaly, congestion, pleural effusion, pneumonic infiltration and atelectasis, to each side (where applicable). The non-private AI models were compared with privacy-preserving (DP) models with respect to privacy-utility trade-offs (measured as area under the receiver-operator-characteristic curve (AUROC)), and privacy-fairness trade-offs (measured as Pearson-R or Statistical Parity Difference). The non-private AI model achieved an average AUROC score of 0.90 over all labels, whereas the DP AI model with a privacy budget of epsilon=7.89 resulted in an AUROC of 0.87, i.e., a mere 2.6% performance decrease compared to non-private training. The privacy-preserving training of diagnostic AI models can achieve high performance with a small penalty on model accuracy and does not amplify discrimination against age, sex or co-morbidity. We thus encourage practitioners to integrate state-of-the-art privacy-preserving techniques into medical AI model development.