The channel hardening effect is less pronounced in the cell-free massive multiple-input multiple-output (mMIMO) system compared to its cellular counterpart, making it necessary to estimate the downlink effective channel gains to ensure decent performance. However, the downlink training inadvertently creates an opportunity for adversarial nodes to launch pilot spoofing attacks (PSAs). First, we demonstrate that adversarial distributed access points (APs) can severely degrade the achievable downlink rate. They achieve this by estimating their channels to users in the uplink training phase and then precoding and sending the same pilot sequences as those used by legitimate APs during the downlink training phase. Then, the impact of the downlink PSA is investigated by rigorously deriving a closed-form expression of the per-user achievable downlink rate. By employing the min-max criterion to optimize the power allocation coefficients, the maximum per-user achievable rate of downlink transmission is minimized from the perspective of adversarial APs. As an alternative to the downlink PSA, adversarial APs may opt to precode random interference during the downlink data transmission phase in order to disrupt legitimate communications. In this scenario, the achievable downlink rate is derived, and then power optimization algorithms are also developed. We present numerical results to showcase the detrimental impact of the downlink PSA and compare the effects of these two types of attacks.