  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.
