I have seen film of legs being cut off live frogs, so that part is true, at least. It was being done in Asia, I think for export to Europe. The frogs were caught from the wild, and I know this leads to pest problems in the paddy fields.
In fact, some amphibians (particularly newts) can regrow limbs -- but unfortunately not those frogs. The frogs were being cut above the "hips", so they lost the whole of the legs and presumably part of the pelvis. There is no doubt that they die, although they may survive for a few days.
Re-growing of limbs in amphibians only happens, I think, when a good bit of the limb is still present. It even happens a little in humans -- if a very young baby loses the tip of a finger below the last joint, it often regrows without any scar or missing bits (fingerprints and nails and all). Other injuries often don't scar in young children either.
Apparently, the loss of re-growing in adult mammals is to do with the need for a warm-blooded animal to heal very fast, so scarring is favoured over re-growing. Recent research is looking into ways of suppressing scarring in the hope of persuading human limbs to regrow.