  This paper considers an $N$-player stochastic Nash game in which the $i$th
player minimizes a composite objective $f_i(x) + r_i(x_i)$, where $f_i$ is
expectation-valued and $r_i$ has an efficient prox-evaluation. In this context,
we make the following contributions. (i) Under a strong monotonicity assumption
on the concatenated gradient map, we derive ({\bf optimal}) rate statements and
oracle complexity bounds for the proposed variable sample-size proximal
stochastic gradient-response (VS-PGR) scheme; (ii) We overlay (VS-PGR) with a
consensus phase with a view towards developing distributed protocols for
aggregative stochastic Nash games. Notably, when the sample-size and the number
of consensus steps at each iteration grow at a suitable rate, a linear rate of
convergence can be achieved; (iii) Finally, under a suitable contractive
property associated with the proximal best-response (BR) map, we design a
variable sample-size proximal BR (VS-PBR) scheme, where the proximal BR is
computed by solving a sample-average problem. If the batch-size for computing
the sample-average is raised at a suitable rate, we show that the resulting
iterates converge at a linear rate and derive the oracle complexity.
