We study the problem of uplink compression for cell-free multi-input multi-output networks with limited fronthaul capacity. In compress-forward mode, remote radio heads (RRHs) compress the received signal and forward it to a central unit for joint processing. While previous work has focused on a transform-based approach, which optimizes the transform matrix that reduces signals of high dimension to a static pre-determined lower dimension, we propose a rate-based approach that simultaneously finds both dimension and compression adaptively. Our approach accommodates for changes to network traffic and fronthaul limits. Using mutual information as the objective, we obtain the theoretical network capacity for adaptive compression and decouple the expression to enable decentralization. Furthermore, using channel statistics and user traffic density, we show different approaches to compute an efficient representation of side information that summarizes global channel state information and is shared with RRHs to assist compression. While keeping the information exchange overhead low, our decentralized implementation of adaptive compression shows competitive overall network performance compared to a centralized approach.