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Well the gravitational waves come from a mass that is moving. It's like electromagnetic waves are created by a moving charge. But because gravity is so weak you need it to be a very big mass moving very fast to be detectable. When black holes merge they spiral in and at the last moment they get to extremely high RPM with all that mass moving very fast.
Kind of an interesting thought but I don't know if it really counts to say that the mass and location of the black hole is really "information from inside the horizon" even though technically the center of mass is inside the horizon.
Although we're solidly in the realm of fantasy thought experiment, it struck me that - in principal - if one was inside the black hole, with sufficient mass under your control, you could pass a signal to outside the hole by shifting the mass this way or that.
Obviously we're taking vanishingly small windows of time. But in principal it seems that you could react to something inside the horizon, exert your will on the movement of something super massive, and that be detectable to someone outside the horizon?
Wouldn't shifting the mass require pushing against another mass? In that case, might those two signals cancel each other out?