This chapter concentrates on a class of distributed convex optimization problems subject to local feasible sets, equality and inequality constraints, where the global objective function consists of a sum of locally smooth convex functions and nonsmooth regularization terms. To address this problem, a synchronous full-distributed primal-dual proximal splitting algorithm (Syn-FdPdPs) is presented, which avoids the unapproximable property of the proximal operator with respect to inequality constraints via logarithmic barrier functions. Following the proposed distributed protocol, each agent carries out local information exchange without any global coordination and weight balancing strategies introduced in most consensus algorithms. In addition, a randomized version of the proposed algorithm (Rand-FdPdPs) is conducted through subsets of activated agents, which further removes the global clock coordinator. Theoretically, with the help of asymmetric forward-backward-adjoint (AFBA) splitting technique, the convergence results of the proposed algorithms are provided under the same local step size conditions. Finally, the effectiveness and practicability of the proposed algorithms are demonstrated by numerical simulations on the least-square and least absolute deviation problems.