Do lifting magnets share the weight of objects they are attracting? I've done some research and people have said the object the magnet is lifting up against gravity is exerting the same magnitude of lifting force downwards on the lifting magnet.

If whatever is supporting the magnet experiences both the magnet and object's weights combined, why is that so? Shouldn't the force, assuming it is magnetic, exerted on the object to the lifting magnet be weaker than the lifting magnetic force as the object has a lower strength of B (magnetic field) than the B of the lifting magnet?
 A: Remember that forces act on both objects being attraced to each other, by Newton's third law. So you can think that, when the magnet is pulling on the object, the object is pulling on the magnet.
First, the object is not accelerating, so the net force on it must be zero. It is being pulled down by gravity (due to its weight) and pulled up by the magnet, so the magnetic attraction strength is equal to the weight of the object.
Next, the magnet is not accelerating, so the net force on it must be zero. It is being pulled down by gravity (i.e., due to the weight of the magnet itself), and by the magnetic attraction from the object, equal to the weight of the object. The force your hand must provide to counter these two forces is thus equal in magnitude to the weight of the magnet plus the strength of the magnetic pull from the object, which in total equals the weight of the magnet plus the weight of the object.
This is intuitive because if there was no magnetic attraction but just one big sticky lump of stuff, you'd have to support all of its weight in order to make it not accelerate downward.
In fact, this is entirely equivalent to supporting an object dangling from a string, where the magnetic force is then replaced by the tension force in the rope.
