The emergence of smartphones and internet accessibility around the globe have enabled billions of people to be connected to the digital world. Due to the popularity of instant messaging applications and social media, a large quantity of personal data is in text format, and processing text data in a privacy-preserving manner poses unique challenges. While existing reviews focus on privacy concerns from specific algorithmic perspectives or target only a particular domain, such as healthcare or smart metering, they fail to provide a comprehensive view that addresses the multi-layered privacy risks inherent to text data processing. Existing works often limit their scope to specialized solutions like differential privacy, anonymization, or federated learning, neglecting a broader spectrum of challenges. To fill this gap, we present a comprehensive review of privacy-enhancing solutions for text data processing in the present literature and classify the works into six categories of privacy risks: (i) unintentional memorability, (ii) membership inference, (iii) exposure and re-identification, (iv) language models and word embeddings, (v) authorship attribution, and (vi) collaborative processing. We then analyze existing privacy-enhancing solutions for text data by considering the aforementioned privacy risks. Finally, we identified several research gaps, including the need for comprehensive privacy metrics, explainable algorithms, and privacy in social media analytics.