Abstract: Actor-critic (AC) algorithms have been widely used in decentralized multi-agent systems to learn the optimal joint control policy. However, existing decentralized AC algorithms either need to share agents' sensitive information or lack communication-efficiency. In this work, we develop decentralized AC and natural AC (NAC) algorithms that avoid sharing agents' local information and are sample and communication-efficient. In both algorithms, agents share only noisy rewards and use mini-batch local policy gradient updates to ensure high sample and communication efficiency. Particularly for decentralized NAC, we develop a decentralized Markovian SGD algorithm with an adaptive mini-batch size to efficiently compute the natural policy gradient. Under Markovian sampling and linear function approximation, we prove that the proposed decentralized AC and NAC algorithms achieve the state-of-the-art sample complexities $\mathcal{O}(\epsilon^{-2}\ln\epsilon^{-1})$ and $\mathcal{O}(\epsilon^{-3}\ln\epsilon^{-1})$, respectively, and achieve an improved communication complexity $\mathcal{O}(\epsilon^{-1}\ln\epsilon^{-1})$. Numerical experiments demonstrate that the proposed algorithms achieve lower sample and communication complexities than the existing decentralized AC algorithms.