It has been shown in the field of Author Profiling that texts may inadvertently reveal sensitive information about their authors, such as gender or age. This raises important privacy concerns that have been extensively addressed in the literature, in particular with the development of methods to hide such information. We argue that, when these texts are in fact messages exchanged between individuals, this is not the end of the story. Indeed, in this case, a second party, the intended recipient, is also involved and should be considered. In this work, we investigate the potential privacy leaks affecting them, that is we propose and address the problem of Recipient Profiling. We provide empirical evidence that such a task is feasible on several publicly accessible datasets (https://huggingface.co/datasets/sileod/recipient_profiling). Furthermore, we show that the learned models can be transferred to other datasets, albeit with a loss in accuracy.