Due to the structural and economic features of steel–concrete–steel (SCS) structural systems compared with conventional reinforced concrete ones, they are now used for a range of structural applications. Rubcrete, in which crumbed rubber from scrap tires partially replaces mineral aggregates in concrete, can be used instead of conventional concrete. Utilizing rubber waste in concrete potentially results in a more ductile lightweight concrete that can introduce additional features to the SCS structural members. This study aimed to explore different concrete core materials in SCS beams and the appropriate shear connectors required. In this study, four SCS sandwich beams were tested experimentally under incrementally increasing flexure cyclic loading. Each beam had a length of 1000 mm, and upper and lower steel plates with 3 mm thickness sandwiched the concrete core, which had a cross-section of 150 mm × 150 mm. Two of the beams were constructed out of Rubcrete core with welded and bolted shear connectors, while the other two beams were constructed with welded shear connectors and either conventional concrete or lightweight expanded clay aggregate (LECA) concrete cores. The performance of the SCS sandwich beams including damage pattern, failure mode, load-displacement response, and energy dissipation behavior was compared. The results showed that, while Rubcrete was able to provide similar concrete cracking behavior and strength to that of conventional concrete, LECA concrete degraded the strength properties of SCS. Using bolted shear connectors instead of welded ones caused a high number of cracks that resulted in a reduced ductility and deflection capacity of the beam before failure. The rubberized concrete specimen presented an improved ductility and deflection capacity compared with its conventional concrete counterpart.