A new architecture is provided for the efficient generation of shaped beams and multiple shaped beams (in amplitude, phase, and polarization), with leaky-wave (LW) metasurfaces (MTSs). Those antennas are usually fed by a surface-wave (SW) launcher, which evenly illuminates the MTS versus azimuth, through a cylindrical SW. When the interest lies in generating pencil beams or particular types of shaped beams, the required radiating aperture field (linked to the Fourier transform of the desired radiation pattern) on the MTS is almost azimuthally symmetric in amplitude. However, for a wide class of shaped beams, the desired aperture field is far from being azimuthally symmetric. This leads to an MTS design in which some sectors exhibit a very small surface impedance modulation. Those sectors tend to leave the SW intact until it reaches the rim of the surface, where the power is either reflected or diffracted at the slab rim. We solve this efficiency problem by reducing the amount of power fed into some sectors, through the use of an array of feeding monopoles. The azimuthal illumination of the feeder should be designed to match the azimuthal asymmetry of the desired aperture field. The corresponding MTS surface impedance is then determined using a method of moments (MoM)-based synthesis technique. Several examples are provided to illustrate the concept. The improvement in terms of antenna efficiency is numerically demonstrated.