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Deep segmentation neural networks require large training datasets with pixel-wise segmentations, which are expensive to obtain in practice. Mixed supervision could mitigate this difficulty, with a small fraction of the data containing complete pixel-wise annotations, while the rest being less supervised, e.g., only a handful of pixels are labeled. In this work, we propose a dual-branch architecture, where the upper branch (teacher) receives strong annotations, while the bottom one (student) is driven by limited supervision and guided by the upper branch. In conjunction with a standard cross-entropy over the labeled pixels, our novel formulation integrates two important terms: (i) a Shannon entropy loss defined over the less-supervised images, which encourages confident student predictions at the bottom branch; and (ii) a Kullback-Leibler (KL) divergence, which transfers the knowledge from the predictions generated by the strongly supervised branch to the less-supervised branch, and guides the entropy (student-confidence) term to avoid trivial solutions. Very interestingly, we show that the synergy between the entropy and KL divergence yields substantial improvements in performances. Furthermore, we discuss an interesting link between Shannon-entropy minimization and standard pseudo-mask generation and argue that the former should be preferred over the latter for leveraging information from unlabeled pixels. Through a series of quantitative and qualitative experiments, we show the effectiveness of the proposed formulation in segmenting the left-ventricle endocardium in MRI images. We demonstrate that our method significantly outperforms other strategies to tackle semantic segmentation within a mixed-supervision framework. More interestingly, and in line with recent observations in classification, we show that the branch trained with reduced supervision largely outperforms the teacher.