We propose a method for separating and detecting the nontensor modes of stochastic gravitational-wave backgrounds (SGWBs) using networks of space-based gravitational-wave detectors. We consider four distinct data-reconstruction schemes for the coinclination and anti-inclination orbital configurations of the LISA-Taiji network. We find that the coinclination configuration offers its advantages over the anti-inclination one and can achieve signal-to-noise ratios up to 17.3 for the vector modes and 10.4 for the scalar modes with the energy density spectrum as Ω𝑝GW⁡(𝑓)=10−12. Our method can be used to measure beyond-general-relativity polarization modes of SGWBs at the mHz frequency band, opening a new avenue for testing alternative gravity theories.