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Due to the drastic increase of electricity prosumers, i.e., energy consumers that are also producers, smart grids have become a key solution for electricity infrastructure. In smart grids, one of the most crucial requirements is the privacy of the final users. The vast majority of the literature addresses the privacy issue by providing ways of hiding user's electricity consumption. However, open issues in the literature related to the privacy of the electricity producers still remain. In this paper, we propose a framework that preserves the secrecy of prosumers' identities and provides protection against the traffic analysis attack in a competitive market for energy trade in a Neighborhood Area Network (NAN). In addition, the amount of bidders and of successful bids are hidden from malicious attackers by our framework. Due to the need for small data throughput for the bidders, the communication links of our framework are based on a proprietary communication system. Still, in terms of data security, we adopt the Advanced Encryption Standard (AES) 128 bit with Exclusive-OR (XOR) keys due to their reduced computational complexity, allowing fast processing. Our framework outperforms the state-of-the-art solutions in terms of privacy protection and trading flexibility in a prosumer-to-prosumer design.