Optical biosensors target widespread applications, such as drug discovery, medical diagnostics, food quality control, and environmental monitoring. Here, we propose a novel plasmonic biosensor on the end-facet of a dual-core single-mode optical fiber. The concept uses slanted metal gratings on each core, interconnected by a metal stripe biosensing waveguide to couple the cores via the propagation of surface plasmons along the end facet. The scheme enables operation in transmission (core-to-core), thereby eliminating the need to separate the reflected light from the incident light. Importantly, this simplifies and reduces the cost of the interrogation setup because a broadband polarization-maintaining optical fiber coupler or circulator is not required. The proposed biosensor enables remote sensing because the interrogation optoelectronics can be located remotely. In vivo biosensing and brain studies are also enabled because the end-facet can be inserted into a living body, once properly packaged. It can also be dipped into a vial, precluding the need for microfluidic channels or pumps. Bulk sensitivities of 880 nm/RIU and surface sensitivities of 1 nm/nm are predicted under spectral interrogation using cross-correlation analysis. The configuration is embodied by robust and experimentally realizable designs that can be fabricated, e.g., using metal evaporation and focused ion beam milling.