Biased decisions made by automatic systems have led to growing concerns in research communities. Recent work from the NLP community focuses on building systems that make fair decisions based on text. Instead of relying on unknown decision systems or human decision-makers, we argue that a better way to protect data providers is to remove the trails of sensitive information before publishing the data. In light of this, we propose a new privacy-aware text rewriting task and explore two privacy-aware back-translation methods for the task, based on adversarial training and approximate fairness risk. Our extensive experiments on three real-world datasets with varying demographical attributes show that our methods are effective in obfuscating sensitive attributes. We have also observed that the fairness risk method retains better semantics and fluency, while the adversarial training method tends to leak less sensitive information.