Emerging applications in multi-agent environments such as internet-of-things, networked sensing, autonomous systems and federated learning, call for decentralized algorithms for finite-sum optimizations that are resource-efficient in terms of both computation and communication. In this paper, we con- sider the prototypical setting where the agents work collaboratively to minimize the sum of local loss functions by only communicating with their neighbors over a predetermined network topology. We develop a new algorithm, called DEcentralized STochastic REcurSive gradient methodS (DESTRESS) for nonconvex finite-sum optimization, which matches the near-optimal incremental first-order oracle (IFO) complexity of state-of-the-art centralized algorithms for finding first-order stationary points, and significantly improves over existing decentralized algorithms. The communication complexity of DESTRESS also improves upon prior arts over a wide range of parameter regimes. DESTRESS leverages several key algorithm design ideas including stochastic recursive gradient updates with mini-batches for local computation, gradient tracking with extra mixing for per-iteration communication, together with careful choices of hyper-parameters and new analysis frameworks to provably achieve a desirable computation-communication trade-off.