The interaction between a single hole and a two-dimensional, paramagnetic, homogeneous electron gas is studied using diffusion quantum Monte Carlo simulations. Electron-hole relaxation energies, pair-correlation functions, and electron-hole center-of-mass momentum densities are reported for a range of electron-hole mass ratios and electron densities. We find numerical evidence of a crossover from a collective excitonic state to a trion-dominated state in a density range in agreement with that found in recent experiments on quantum-well heterostructures.