Modern society depends on the flow of information over online social networks, and users of popular platforms generate significant behavioral data about themselves and their social ties. However, it remains unclear what fundamental limits exist when using these data to predict the activities and interests of individuals, and to what accuracy such predictions can be made using an individual's social ties. Here we show that 95% of the potential predictive accuracy for an individual is achievable using their social ties only, without requiring that individual's data. We use information theoretic tools to estimate the predictive information within the writings of Twitter users, providing an upper bound on the available predictive information that holds for any predictive or machine learning methods. As few as 8-9 of an individual's contacts are sufficient to obtain predictability comparable to that of the individual alone. Distinct temporal and social effects are visible by measuring information flow along social ties, allowing us to better study the dynamics of online activity. Our results have distinct privacy implications: information is so strongly embedded in a social network that in principle one can profile an individual from their available social ties even when the individual forgoes the platform completely.