  Interactions between quasiparticles are of fundamental importance and
ultimately determine the macroscopic properties of quantum matter. A famous
example is the phenomenon of superconductivity, which arises from attractive
electron-electron interactions that are mediated by phonons or even other more
exotic fluctuations in the material. Here we introduce mobile exciton
impurities into a two-dimensional electron gas and investigate the interactions
between the resulting Fermi polaron quasiparticles. We employ multi-dimensional
coherent spectroscopy on monolayer WS$_2$, which provides an ideal platform for
determining the nature of polaron-polaron interactions due to the underlying
trion fine structure and the valley specific optical selection rules. At low
electron doping densities, we find that the dominant interactions are between
polaron states that are dressed by the same Fermi sea. In the absence of bound
polaron pairs (bipolarons), we show using a minimal microscopic model that
these interactions originate from a phase-space filling effect, where excitons
compete for the same electrons. We furthermore reveal the existence of a
bipolaron bound state with remarkably large binding energy, involving excitons
in different valleys cooperatively bound to the same electron. Our work lays
the foundation for probing and understanding strong electron correlation
effects in two-dimensional layered structures such as moir\'e superlattices.
