We define auto- and cross-correlation functions capable of capturing dynamical characteristics induced by local phase-space structures in a general dynamical system. These correlation functions are calculated in the standard map for a range of values of the nonlinearity parameter k. Using a model of noninteracting particles, each evolving according to the same standard map dynamics and located initially at specific phase-space regions, we show that for 0.6<k≤1.2 long-range cross correlations emerge. They occur as an ensemble property of particle trajectories by an appropriate choice of the phase-space cells used in the statistical averaging. In this region of k values the single-particle phase space is either dominated by local chaos (k≤k(c) with k(c)≈0.97) or it is characterized by the transition from local to global chaos (k(c)<k≤1.2). Introducing suitable symbolic dynamics we demonstrate that the emergence of long-range cross correlations can be attributed to the existence of an effective intermittent dynamics in specific regions of the phase space. Our findings support the recently established relation of intermittent dynamics and cross correlations [F. K. Diakonos, A. K. Karlis, and P. Schmelcher, Europhys. Lett. 105, 26004 (2014)] in simple one-dimensional intermittent maps, suggesting its validity also for two-dimensional Hamiltonian maps.