The fine-grained localization of clinicians in the operating room (OR) is a key component to design the new generation of OR support systems. Computer vision models for person pixel-based segmentation and body-keypoints detection are needed to better understand the clinical activities and the spatial layout of the OR. This is challenging, not only because OR images are very different from traditional vision datasets, but also because data and annotations are hard to collect and generate in the OR due to privacy concerns. To address these concerns, we first study how joint person pose estimation and instance segmentation can be performed on low resolutions images from 1x to 12x. Second, to address the domain shift and the lack of annotations, we propose a novel unsupervised domain adaptation method, called \emph{AdaptOR}, to adapt a model from an \emph{in-the-wild} labeled source domain to a statistically different unlabeled target domain. We propose to exploit explicit geometric constraints on the different augmentations of the unlabeled target domain image to generate accurate pseudo labels, and using these pseudo labels to train the model on high- and low-resolution OR images in a \emph{self-training} framework. Furthermore, we propose \emph{disentangled feature normalization} to handle the statistically different source and target domain data. Extensive experimental results with detailed ablation studies on the two OR datasets \emph{MVOR+} and \emph{TUM-OR-test} show the effectiveness of our approach against strongly constructed baselines, especially on the low-resolution privacy-preserving OR images. Finally, we show the generality of our method as a semi-supervised learning (SSL) method on the large-scale \emph{COCO} dataset, where we achieve comparable results with as few as \textbf{1\%} of labeled supervision against a model trained with 100\% labeled supervision.