This paper investigates a class of distributed optimization problems where the objective function is given by the sum of twice differentiable convex functions and a convex non-differentiable part. The setting assumes a network of communicating agents in which each individual agent's objective is captured by a summand of the aggregate objective function, and agents cooperate through an information exchange with their neighbors. We devise a second order method by transforming the problem into a continuously differentiable form using proximal operators, and truncating the Taylor expansion of the Hessian inverse so that a distributed implementation of the algorithm is possible. We prove global linear convergence (without backtracking), under usual strong convexity assumptions, and further demonstrate the effectiveness of our scheme through numerical simulations.