Federated learning (FL) enables collaborative model training without centralizing data, but exchanging high-dimensional updates can expose sensitive information and incur substantial communication costs. We present FedRP, a communication-efficient method combining Gaussian random projection with consensus optimization based on the alternating direction method of multipliers (ADMM). In each round, clients project their model parameters into an $m$-dimensional space using a shared random matrix hidden from the server, which aggregates only compressed representations. We establish a high-probability guarantee linking projected-space consensus to proximity among client models and derive an $(ε,δ)$-differential privacy guarantee for each release under bounded $\ell_2$-sensitivity and a positive lower bound on parameter norms. With randomized parameter vectors and the projection matrix also limits information available to common reconstruction attacks. Experiments on MNIST and CIFAR-10 with LeNet-5 and a custom convolutional network show that FedRP achieves accuracy comparable to FedAvg and consistently exposes noise-perturbed privacy via FedAvg. Because clients transmit $m$ rather than $n$ values per round, FedRP reduces communication by orders of magnitude when $m \ll n$. The results demonstrate a favorable trade-off among accuracy, privacy, and communication efficiency. Code is available at https://github.com/mhnarimani/FedRP

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