Visual grounding is a task that seeks to predict the specific location of an object or region described by a linguistic expression within an image. Despite the recent success, existing methods still suffer from two problems. First, most methods use independently pre-trained unimodal feature encoders for extracting expressive feature embeddings, thus resulting in a significant semantic gap between unimodal embeddings and limiting the effective interaction of visual-linguistic contexts. Second, existing attention-based approaches equipped with the global receptive field have a tendency to neglect the local information present in the images. This limitation restricts the semantic understanding required to distinguish between referred objects and the background, consequently leading to inadequate localization performance. Inspired by the recent advance in knowledge distillation, in this paper, we propose a DUal knowlEdge disTillation (DUET) method for visual grounding models to bridge the cross-modal semantic gap and improve localization performance simultaneously. Specifically, we utilize the CLIP model as the teacher model to transfer the semantic knowledge to a student model, in which the vision and language modalities are linked into a unified embedding space. Besides, we design a self-distillation method for the student model to acquire localization knowledge by performing the region-level contrastive learning to make the predicted region close to the positive samples. To this end, this work further proposes a Semantics-Location Aware sampling mechanism to generate high-quality self-distillation samples. Extensive experiments on five datasets and ablation studies demonstrate the state-of-the-art performance of DUET and its orthogonality with different student models, thereby making DUET adaptable to a wide range of visual grounding architectures. Code are available on DUET.