Semi-supervised learning has proven highly effective in tackling the challenge of limited labeled training data in medical image segmentation. In general, current approaches, which rely on intra-image pixel-wise consistency training via pseudo-labeling, overlook the consistency at more comprehensive semantic levels (e.g., object region) and suffer from severe discrepancy of extracted features resulting from an imbalanced number of labeled and unlabeled data. To overcome these limitations, we present a new \underline{Du}al \underline{C}ross-\underline{i}mage \underline{S}emantic \underline{C}onsistency (DuCiSC) learning framework, for semi-supervised medical image segmentation. Concretely, beyond enforcing pixel-wise semantic consistency, DuCiSC proposes dual paradigms to encourage region-level semantic consistency across: 1) labeled and unlabeled images; and 2) labeled and fused images, by explicitly aligning their prototypes. Relying on the dual paradigms, DuCiSC can effectively establish consistent cross-image semantics via prototype representations, thereby addressing the feature discrepancy issue. Moreover, we devise a novel self-aware confidence estimation strategy to accurately select reliable pseudo labels, allowing for exploiting the training dynamics of unlabeled data. Our DuCiSC method is extensively validated on four datasets, including two popular binary benchmarks in segmenting the left atrium and pancreas, a multi-class Automatic Cardiac Diagnosis Challenge dataset, and a challenging scenario of segmenting the inferior alveolar nerve that features complicated anatomical structures, showing superior segmentation results over previous state-of-the-art approaches. Our code is publicly available at \href{https://github.com/ShanghaiTech-IMPACT/DuCiSC}{https://github.com/ShanghaiTech-IMPACT/DuCiSC}.