Why not? If I see a crime being committed, should I be allowed to report it to police without a warrant? There's no difference with my private camera, it's even better cause the barrier for false/mistaken reporting is higher.
Private doorbell cameras (and most flock cameras for that matter) point at mostly, or entirely, public space. Or in case of the former also my own property.
Frankly if I sit on my porch and see thru the window across that my neighbor is beating his wife, should I need a warrant to report it? That sounds ludicrous to me.