This article considers the distributed structured optimization problem of collaboratively minimizing the global objective function composed of the sum of local cost 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, we derive the synchronous distributed primal–dual splitting (S-DPDS) algorithm with uncoordinated stepsizes. Meanwhile, we develop the asynchronous version of the algorithm in light of the randomized block-coordinate method (A-DPDS). 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, we demonstrate the efficiency of S-DPDS and A-DPDS algorithms by numerical experiments.