This chapter considers the distributed structured optimization problem of collaboratively minimizing the global objective function composed of the sum of local objective functions. Each local objective function involves a Lipschitz-differentiable convex function, a nonsmooth convex function, and a linear composite nonsmooth convex function. For such problems, a synchronous distributed primal-dual splitting (S-DPDS) algorithm is developed with uncoordinated step sizes. Then, its asynchronous version in light of the randomized block-coordinate method (A-DPDS) is proposed. Further, the convergence results show the relaxed range and concise form of the acceptable parameters, which indicates that the algorithms are conducive to the selection of parameters in practical applications. Finally, the efficiency of S-DPDS and A-DPDS algorithms is testified by numerical experiments.