Proximity effects in two-dimensional (2D) van der Waals heterostructures offer controllable ways to tailor the electronic band structure of adjacent materials. Proximity exchange in particular is important for making materials magnetic without hosting magnetic ions. Such synthetic magnets could be used for studying magnetotransport in high-mobility 2D materials, or magneto-optics in highly absorptive nominally nonmagnetic semiconductors. Using first-principles calculations, we show that the proximity exchange in monolayer MoSe2 and WSe2 due to ferromagnetic monolayer CrI3 can be tuned (even qualitatively) by twisting and gating. Remarkably, the proximity exchange remains the same when using antiferromagnetic CrI3 bilayer, paving the way for optical and electrical detection of layered antiferromagnets. Interestingly, the proximity exchange is opposite to the exchange of the adjacent antiferromagnetic layer. Finally, we show that the proximity exchange is confined to the layer adjacent to CrI3, and that adding a separating hBN barrier drastically reduces the proximity effect. We complement our ab initio results with tight-binding modeling and solve the Bethe-Salpeter equation to provide experimentally verifiable optical signatures (in the exciton spectra) of the proximity exchange effects.