Doxing refers to the practice of disclosing sensitive personal information about a person without their consent. This form of cyberbullying is an unpleasant and sometimes dangerous phenomenon for online social networks. Although prior work exists on automated identification of other types of cyberbullying, a need exists for methods capable of detecting doxing on Twitter specifically. We propose and evaluate a set of approaches for automatically detecting second- and third-party disclosures on Twitter of sensitive private information, a subset of which constitutes doxing. We summarize our findings of common intentions behind doxing episodes and compare nine different approaches for automated detection based on string-matching and one-hot encoded heuristics, as well as word and contextualized string embedding representations of tweets. We identify an approach providing 96.86% accuracy and 97.37% recall using contextualized string embeddings and conclude by discussing the practicality of our proposed methods.