Compared to monolayer graphene, electrons in Bernal-stacked bilayer graphene (BLG) have an additional layer degree of freedom, offering a platform for developing layered spintronics with the help of proximity-induced magnetism. Based on an effective phenomenological model, we systematically study the effect of this magnetism on the spin-dependent band structure near the Fermi energy and identify the magnetic phases induced in BLG by proximity with magnets. We show that spin polarization can develop in BLG due to this proximity effect. This spin polarization depends strongly on the layer distribution of magnetism, and can always be controlled by gate voltage which shifts spin-dependent band edges and modifies the total band gap. We further show that the band spin polarization can be modified by the proximity-induced staggered sublattice potential. By taking full advantage of layer-dependent magnetism in BLG, we propose that spintronic devices such as a spin filter, a giant magnetoresistance device, and a spin diode can operate under fully electric control, which is easier than the common magnetic field control.