Massive datasets of human activity are now available, revolutionizing research on human dynamics and computational social science. We study the online posts of thousands of Twitter users and their followers. Information flows across the Twitter network by these posts, but it is unclear how much information flows, how much influence do users have on one another, and if we can accurately measure these quantities.Treating each Twitter user's text stream as a symbolic time series, the entropy rate measures how much information about a future word choice is available in the past history. We apply theorems from data compression to estimate a "correlated" entropy that accounts for both temporal ordering and long-range correlations in the data. The correlated entropy rate estimates the inherent uncertainty about someone's future word choice.Crucially, this technique can also capture social information transfer between pairs of users (denoted egos and alters), via a cross-entropy that estimates how much information about the ego's future word choice is present in the alter's past words. Some alters contain nearly as much information about the ego as the ego itself, but other ego-alter pairs show little information flux. Taken together, these results provide new quantitative bounds on information transfer in social networks, useful for better understanding the spread of ideas and influence in human populations.