  We consider a collaborative iterative algorithm with two agents and a fusion
center for estimation of a real valued function (or ``model") on the set of
real numbers. While the data collected by the agents is private, in every
iteration of the algorithm, the models estimated by the agents are uploaded to
the fusion center, fused, and, subsequently downloaded by the agents. We
consider the estimation spaces at the agents and the fusion center to be
Reproducing Kernel Hilbert Spaces (RKHS). Under suitable assumptions on these
spaces, we prove that the algorithm is consistent, i.e., there exists a
subsequence of the estimated models which converges to a model in the strong
topology. To this end, we define estimation operators for the agents, fusion
center, and, for every iteration of the algorithm constructively. We define
valid input data sequences, study the asymptotic properties of the norm of the
estimation operators, and, find sufficient conditions under which the
estimation operator until any iteration is uniformly bounded. Using these
results, we prove the existence of an estimation operator for the algorithm
which implies the consistency of the considered estimation algorithm.
