Locally resonant metamaterials can achieve unprecedented vibroacoustic attenuation in partition panels by subwavelength distributions of small attached resonators. However, conventional metamaterial panels employ single-mode transversal resonators and induce only narrowband improvements. In this work, we address this limitation by investigating and designing multi-modal metamaterial panels, which exploit multiple translational and rotational modes within a single resonator to achieve broadband attenuation. To achieve efficient modelling, we develop dedicated effective medium representations through equivalent homogenized material properties, tailored to include multiple translational and rotational local resonances. For the idealized case of metamaterial panels with infinite extent, the approach allows for the analytical prediction of the (multiple) bandgaps created for bending waves and of the resulting diffuse field sound transmission loss (STL). For real-world scenarios in which the effects of finite size, boundary conditions, and non-uniform resonator distributions are important, efficient diffuse STL predictions are achieved by Deterministic - Statistical Energy Analysis (Det-SEA). Here an effective medium finite element model of the finite-sized metamaterial panel is coupled with a diffuse model of the surrounding sound fields. An important design question is then addressed, that is how to achieve suitable local resonator layouts that obtain an adequate amount of resonances across the target frequency range. This problem is tackled by developing efficient numerical optimization methodologies that exploit effective medium modelling. The optimization objective is to maximize broadband diffuse STL while constraining the maximum mass of the attached resonators. As a showcase, we target the suppression of the broadband STL dip due to coincidence in orthotropic host plates. At first, promising resonator layouts based on physical insight are proposed and parametrically optimized. Afterwards, non-intuitive designs with an increased number of local resonances are obtained through density-based topology optimization, in which free material distribution is allowed for maximum design freedom. The results of the work open up new possibilities for broadband vibroacoustic attenuation in functional applications through optimized multi-modal metamaterial solutions.