In this paper, we analyze the benefits of including downlink pilots in a cell-free massive MIMO system. We derive an approximate per-user achievable downlink rate for conjugate beamforming processing, which takes into account both uplink and downlink channel estimation errors, and power control. A performance comparison is carried out, in terms of per-user net throughput, considering cell-free massive MIMO operation with and without downlink training, for different network densities. We take also into account the performance improvement provided by max-min fairness power control in the downlink. Numerical results show that, exploiting downlink pilots, the performance can be considerably improved in low density networks over the conventional scheme where the users rely on statistical channel knowledge only. In high density networks, performance improvements are moderate.