We explore the feasibility of automatically finding accounts that publish sensitive content on Twitter. One natural approach to this problem is to first create a list of sensitive keywords, and then identify Twitter accounts that use these words in their tweets. But such an approach may overlook sensitive accounts that are not covered by the subjective choice of keywords. In this paper, we instead explore finding sensitive accounts by examining the percentage of anonymous and identifiable followers the accounts have. This approach is motivated by an earlier study showing that sensitive accounts typically have a large percentage of anonymous followers and a small percentage of identifiable followers. To this end, we first considered the problem of automatically determining if a Twitter account is anonymous or identifiable. We find that simple techniques, such as checking for name-list membership, perform poorly. We designed a machine learning classifier that classifies accounts as anonymous or identifiable. We then classified an account as sensitive based on the percentages of anonymous and identifiable followers the account has. We applied our approach to approximately 100,000 accounts with 404 million active followers. The approach uncovered accounts that were sensitive for a diverse number of reasons. These accounts span across varied themes, including those that are not commonly proposed as sensitive or those that relate to socially stigmatized topics. To validate our approach, we applied Latent Dirichlet Allocation (LDA) topic analysis to the tweets in the detected sensitive and non-sensitive accounts. LDA showed that the sensitive and non-sensitive accounts obtained from the methodology are tweeting about distinctly different topics. Our results show that it is indeed possible to objectively identify sensitive accounts at the scale of Twitter.