We introduce the Red-Blue Unshared Domination (RBUD) problem, a variant of domination problems in graphs. Given an undirected graph G with vertices colored either red or blue, the goal is to find a minimum-size subset of vertices S such that for every red-blue pair of vertices outside S, there exist two distinct vertices in S (called unshared dominators), each dominating one vertex of the pair but not the other. Specifically, for each red vertex \(r \not \in S\) and blue vertex \(b\not \in S\) , there must be a vertex in S that dominates r but not b and another vertex in S that dominates b but not r. We prove that the RBUD problem is NP-hard, even for bipartite planar subcubic graphs with large girth. We also provide an approximation algorithm for general graphs and exact efficient algorithms for trees and graphs with bounded neighborhood diversity. Additionally, we extend our algorithmic approach to graphs with bounded modular-width.