Two-phase viscous gravity current theory has numerous applications in the natural sciences, from small-scale lava, sedimentary and glacial flows, to large-scale flows of partially molten mantle. We develop the general equations for two-phase viscous gravity currents composed of a high viscosity matrix and low viscosity fluid for both constant volume and constant flux conditions. A loss of fluid phase is taken into account at the current's upper boundary and corresponds to the degassing of a lava flow or loss of water in sedimentary flows. As the current spreads, its surface increases and fluid loss is facilitated, which modifies the mixture density and viscosity and thus the current's shape; hence spreading of the flow affects fluid loss and vice-versa. Our results show that two-phase gravity currents retain and transport the fluid out to large distances, but the fluid is almost entirely lost within a region of finite radius. This 'loss radius' depends on the flow's volume or flux, fluid and matrix properties as well as on the size of fluid parcels or matrix permeability. Application to lava flows shows that degassing occurs over a large area, which affects gas release and transport in the atmosphere.