Most existing robotic surgery systems adopt a human-in-the-loop paradigm, often with the surgeon directly teleoperating the robotic system. Adding intelligence to these robots would enable higher-level control, such as supervised autonomy or even full autonomy. However, artificial intelligence (AI) requires large amounts of training data, which is currently lacking. This work proposes SurgSync, a multi-modal data collection framework with offline and online synchronization to support training and real-time inference, respectively. The framework is implemented on a da Vinci Research Kit (dVRK) and introduces (1) dual-mode (online/offline-matching) synchronized recorders, (2) a modern stereo endoscope to achieve image quality on par with clinical systems, and (3) additional sensors such as a side-view camera and a novel capacitive contact sensor to provide ground truth contact data. The framework also incorporates a post-processing toolbox for tasks such as depth estimation, optical flow, and a practical kinematic reprojection method using Gaussian heatmap. User studies with participants of varying skill levels are performed with ex-vivo tissue to provide clinically realistic data, and a network for surgical skill assessment is employed to demonstrate utilization of the collected data. Through the user study experiments, we obtained a dataset of 214 validated instances across multiple canonical training tasks. All software and data are available at surgsync.github.io.