The General Data Protection Regulation (GDPR) was conceived to remove the obstacles to the free movement of personal data while ensuring the protection of natural persons with regard to the processing of such data. The Smart Grid has similar features as any privacy-critical system but, in comparison to the engineering of other architectures, has the peculiarity of being the source of energy consumption data. Electricity consumption constitutes an indirect means to infer personal information. This work looks at the Smart Grid from the perspective of the GDPR, which is especially relevant now given the current growth and diversification of the Smart Grid ecosystem. We provide a review of existing works highlighting the importance of energy consumption as valuable personal data as well as an analysis of the established Smart Grid Architecture Model and its main challenges from a legal viewpoint, in particular the challenge of sharing data with third parties.