While general relativity predicts only two tensor modes for gravitational wave polarization, general metric theories of gravity allows up to four additional modes, including two vector and two scalar modes. Observing the polarization modes of gravitational waves could provide a direct test of the modified gravity. The stochastic gravitational wave background (SGWB), which may be detected by space-based laser-interferometric detectors at design sensitivity, will provide an opportunity to directly measure alternative polarization. In this paper, we investigate the performance of the LISA-TianQin network for detecting alternative polarizations of stochastic backgrounds, and propose a method to separate different polarization modes. First, we generalize the small antenna approximation to compute the overlap reduction functions for SGWB with arbitrary polarization, which is suitable for any time-delay interferometry combination. Then we analyze the detection capability of LISA-TianQin for SGWB with different polarizations. Based on the LISA-TianQin orbital characteristics, we propose a method to distinguish different polarization modes from their mixed data. Compared with ground-based detectors, the LISA-TianQin network is more capable of resolving polarizations of SGWB. In particular, the LISA-TianQin network has the potential to resolve two scalar modes that ground-based detectors cannot.