Electrical impedance tomography (EIT) is a noninvasive imaging modality that allows a continuous assessment of changes in regional bioimpedance of different organs. One of its most common biomedical applications is monitoring regional ventilation distribution in critically ill patients treated in intensive care units. In this work, we put forward a proof-of-principle study that demonstrates how one can reconstruct synchronously measured respiratory or circulatory parameters from the EIT image sequence using a deep learning model trained in an end-to-end fashion. For this purpose, we devise an architecture with a convolutional feature extractor whose output is processed by a recurrent neural network. We demonstrate that one can accurately infer absolute volume, absolute flow, normalized airway pressure and within certain limitations even the normalized arterial blood pressure from the EIT signal alone, in a way that generalizes to unseen patients without prior calibration. As an outlook with direct clinical relevance, we furthermore demonstrate the feasibility of reconstructing the absolute transpulmonary pressure from a combination of EIT and absolute airway pressure, as a way to potentially replace the invasive measurement of esophageal pressure. With these results, we hope to stimulate further studies building on the framework put forward in this work.