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We consider multi-group multicast precoding designs for cell-free massive multiple-input multiple-output (MIMO) systems. To optimize the transmit and receive beamforming strategies, we focus on minimizing the sum of the maximum mean squared errors (MSEs) over the multicast groups, which is then approximated with the sum MSE to simplify the computation and signaling. We adopt an iterative bi-directional training scheme with uplink and downlink precoded pilots to cooperatively design the multi-group multicast precoders at each base station and the combiners at each user equipment in a distributed fashion. An additional group-specific uplink training resource is introduced, which entirely eliminates the need for backhaul signaling for channel state information (CSI) exchange. We also propose a simpler distributed precoding design based solely on group-specific pilots, which can be useful in the case of scarce training resources. Numerical results show that the proposed distributed methods greatly outperform conventional cell-free massive MIMO precoding designs that rely solely on local CSI.