Distributed Nash equilibrium (NE) seeking problems for networked games have been widely investigated in recent years. Despite the increasing attention, communication expenditure is becoming a major bottleneck for scaling up distributed approaches within limited communication bandwidth between agents. To reduce communication cost, an efficient distributed NE seeking (ETC-DNES) algorithm is proposed to obtain an NE for games over directed graphs, where the communication efficiency is improved by event-triggered exchanges of compressed information among neighbors. ETC-DNES saves communication costs in both transmitted bits and rounds of communication. Furthermore, our method only requires the row-stochastic property of the adjacency graph, unlike previous approaches that hinged on double-stochastic communication matrices. We provide convergence guarantees for ETC-DNES on games with restricted strongly monotone mappings, testifying that such a communication method is efficient without sacrificing the accuracy of the algorithm. The algorithm and analysis are extended to a compressed algorithm with stochastic event-triggered mechanism (SETC-DNES). In SETC-DNES, we introduce a random variable in the triggering condition to further enhance algorithm efficiency. We demonstrate that SETC-DNES guarantees linear convergence to the optimal NE while achieving even greater reductions in communication costs compared to ETC-DNES. Finally, numerical simulations illustrate the effectiveness of the proposed algorithms.