We consider the massive MIMO downlink with time-division duplex (TDD) operation and conjugate beamforming transmission. To reliably decode the desired signals, the users need to know the effective channel gain. In this paper, we propose a blind channel estimation method which can be applied at the users and which does not require any downlink pilots. We show that our proposed scheme can substantially outperform the case where each user has only statistical channel knowledge, and that the difference in performance is particularly large in certain types of channel, most notably keyhole channels. Compared to schemes that rely on downlink pilots, our proposed scheme yields more accurate channel estimates for a wide range of signal-to-noise ratios and avoid spending time-frequency resources on pilots.