Collaborative perception allows connected vehicles to exchange sensor information and overcome each vehicle's blind spots. Yet transmitting raw point clouds or full feature maps overwhelms Vehicle-to-Vehicle (V2V) communications, causing latency and scalability problems. We introduce EffiComm, an end-to-end framework that transmits less than 40% of the data required by prior art while maintaining state-of-the-art 3D object detection accuracy. EffiComm operates on Bird's-Eye-View (BEV) feature maps from any modality and applies a two-stage reduction pipeline: (1) Selective Transmission (ST) prunes low-utility regions with a confidence mask; (2) Adaptive Grid Reduction (AGR) uses a Graph Neural Network (GNN) to assign vehicle-specific keep ratios according to role and network load. The remaining features are fused with a soft-gated Mixture-of-Experts (MoE) attention layer, offering greater capacity and specialization for effective feature integration. On the OPV2V benchmark, EffiComm reaches 0.84 mAP@0.7 while sending only an average of approximately 1.5 MB per frame, outperforming previous methods on the accuracy-per-bit curve. These results highlight the value of adaptive, learned communication for scalable Vehicle-to-Everything (V2X) perception.