We study distributed composite optimization over networks: agents minimize a sum of smooth (strongly) convex functions, the agents' sum-utility, plus a nonsmooth (extended-valued) convex one. We propose a general unified algorithmic framework for such a class of problems and provide a unified convergence analysis leveraging the theory of operator splitting. Distinguishing features of our scheme are: (i) When the agents' functions are strongly convex, the algorithm converges at a linear rate, whose dependence on the agents' functions and network topology is decoupled, matching the typical rates of centralized optimization; the rate expression improves on existing results; (ii) When the objective function is convex (but not strongly convex), similar separation as in (i) is established for the coefficient of the proved sublinear rate; (iii) The algorithm can adjust the ratio between the number of communications and computations to achieve a rate (in terms of computations) independent on the network connectivity; and (iv) A by-product of our analysis is a tuning recommendation for several existing (non accelerated) distributed algorithms yielding the fastest provably (worst-case) convergence rate. This is the first time that a general distributed algorithmic framework applicable to composite optimization enjoys all such properties.