Abstract We present an efficient method to solve high-frequency scattering problems by heterogeneous penetrable objects in two dimensions. This is achieved by extending the so-called Local Multiple Traces Formulation, introduced recently by Hiptmair and Jerez-Hanckes, to purely spectral discretizations employing weighted Chebyshev polynomials. Together with regularization strategies to handle boundary integral operators singularities, matrix entries are quickly computed via the Fast Fourier Transform. The resulting Fredholm first-kind formulation is free from spurious resonances, and though ill-conditioned, it possesses built-in Calderón-type preconditioners. Numerical results obtained for different settings validate our claims and greatly motivate future research in this direction.