Kernel design for Gaussian processes (GPs) along with the associated hyper-parameter optimization is a challenging problem. In this paper, we propose a novel grid spectral mixture (GSM) kernel design for GPs that can automatically fit multidimensional data with affordable model complexity and superior modeling capability. To alleviate the curse of dimensionality, we leverage a multicore computing environment to optimize the kernel hyper-parameters in a distributed manner. We further propose a doubly distributed learning algorithm based on the alternating direction method of multipliers (ADMM) which enables multiple agents to learn the kernel hyper-parameters collaboratively. The doubly distributed learning algorithm is shown to be effective in reducing the overall computational complexity while preserving data privacy during the learning process. Experiments on various one-dimensional and multidimensional data sets demonstrate that the proposed kernel design yields superior training and prediction performance compared to its competitors.