We introduce a framework - Artemis - to tackle the problem of learning in a distributed or federated setting with communication constraints. Several workers (randomly sampled) perform the optimization process using a central server to aggregate their computations. To alleviate the communication cost, Artemis allows to compress the information sent in both directions (from the workers to the server and conversely) combined with a memory mechanism. It improves on existing algorithms that only consider unidirectional compression (to the server), or use very strong assumptions on the compression operator. We provide fast rates of convergence (linear up to a threshold) under weak assumptions on the stochastic gradients (noise's variance bounded only at optimal point) in non-i.i.d. setting, highlight the impact of memory for unidirectional and bidirectional compression, and analyze Polyak-Ruppert averaging. We use convergence in distribution to obtain a lower bound of the asymptotic variance that highlights practical limits of compression.