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As the Kepler mission has done for hot exoplanets, the ESA Euclid and NASA Roman missions have the potential to create a breakthrough in our understanding of the demographics of cool exoplanets, including unbound, or "free-floating", planets (FFPs). In this study, we demonstrate the complementarity of the two missions and propose two joint-surveys to better constrain the mass and distance of microlensing events. We first demonstrate that an early brief Euclid survey (7 h) of the Roman microlensing fields will allow the measurement of a large fraction of events relative proper motions and lens magnitudes. Then, we study the potential of simultaneous observations by Roman and Euclid to enable the measurement of the microlensing parallax for the shortest microlensing events. Using detailed simulations of the joint detection yield we show that within one year Roman-Euclid observations will be at least an order of magnitude more sensitive than current ground-based measurements. Depending on the exact distribution of FFP, a joint Roman-Euclid campaign should detect around 130 FFP events within a year, including 110 with measured parallax that strongly constrain the FFP mass, and around 30 FFP events with direct mass and distance measurements. The ability of the joint survey to completely break the microlens mass-distance-velocity degeneracy for a significant subset of events provides a unique opportunity to verify unambiguously the FFP hypothesis or else place abundance limits for FFPs between Earth and Jupiter masses that are up to two orders of magnitude stronger than provided by ground-based surveys. Finally, we study the capabilities of the joint survey to enhance the detection and charcterization of exomoons, and found that it could lead to the detection of the first exomoon.