The structured saddle-point problem involving the infimal convolution in real Hilbert spaces finds applicability in many applied mathematics disciplines. For this purpose, we develop a stochastic primal-dual splitting algorithm with loopless variance-reduction for solving this generic problem. We first prove the weak almost sure convergence of the iterates. We then demonstrate that our algorithm achieves linear convergence in expectation of its iterates as well as convergence of the (smoothed primal-dual and duality) gap function value under the assumption of strong convexity. We also derive the total average complexity and compare it to the most recent advances developed in the available literature.