Stellar-mass binary black hole (BBH) mergers that occur within the disks of active galactic nuclei (AGN) are promising sources for gravitational waves that can be detected by the LIGO, Virgo, and KAGRA interferometers. Some of these events have also been potentially associated with transient electromagnetic flares, indicating that BBH mergers in dense environments may be promising sources of multimessenger signals. To investigate the prospects for electromagnetic emission from these systems, we studied the dynamics of accretion flows onto BBHs that are embedded in AGN disks with numerical simulations. Although recent studies have explored this scenario, they often employed simplified disk models that neglected magnetic fields. We examined how strong magnetic fields affect and regulate the accretion onto these binary systems. In this context, we conducted three-dimensional magnetohydrodynamical local shearing-box simulations of a BBH system embedded within a magnetized disk of an AGN. The dynamically important magnetic fields can drive the formation of well-collimated outflows that can penetrate the vertical extent of the AGN disk. The outflow generation is not ubiquitous, however, and strongly depends on the radial distance of the binary from the supermassive black hole (SMBH). In particular, binaries placed at a larger distance from the central SMBH show a relatively stronger transient accretion and the formation of stronger spiral shocks. Furthermore, the accretion behavior onto the binary system via individual circum-singular disks is also modulated by local AGN disk properties. Our simulations highlight the importance of the shear velocity in the amplification of the toroidal magnetic field component, which plays a crucial role in governing the outflow strength.