In the next wave of swarm-based applications, unmanned aerial vehicles (UAVs) need to communicate with peer drones in any direction of a three-dimensional (3D) space. On a given drone and across drones, various antenna positions and orientations are possible. We know that, in free space, high levels of signal loss are expected if the transmitting and receiving antennas are cross polarized. However, increasing the reflective and scattering objects in the channel between a transmitter and receiver can cause the received polarization to become completely independent from the transmitted polarization, making the cross-polarization of antennas insignificant. Usually, these effects are studied in the context of cellular and terrestrial networks and have not been analyzed when those objects are the actual bodies of the communicating drones that can take different relative directions or move at various elevations. In this work, we show that the body of the drone can affect the received power across various antenna orientations and positions and act as a local scatterer that increases channel depolarization, reducing the cross-polarization discrimination (XPD). To investigate these effects, we perform experimentation that is staged in terms of complexity from a controlled environment of an anechoic chamber with and without drone bodies to in-field environments where drone-mounted antennas are in-flight with various orientations and relative positions with the following outcomes: (i.) drone relative direction can significantly impact the XPD values, (ii.) elevation angle is a critical factor in 3D link performance, (iii.) antenna spacing requirements are altered for co-located cross-polarized antennas, and (iv.) cross-polarized antenna setups more than double spectral efficiency. Our results can serve as a guide for accurately simulating and modeling UAV networks and drone swarms.