Yes, keys are needed, and yes, the spirits use Lucy's MP when she summons them.
However this is not always true.
When Loki was Karen's spirit, he decided to remain in the physical world, and thus kept draining Karen's MP. She tried to summon a second spirit and died because it was too much. So she died, yet Loki stayed in the human world using his own MP. His key wasn't needed anymore, Karen's MP wasn't needed anymore.
When Leo decided to be Gray's partner in Tenrou, Lucy didn't summon him, he came by himself by using his own MP. He even said to Lucy "don't worry, i won't be using your MP, i'll use my own so you can open other gates".
Keys are needed for a summoner to summon, yes, and they drain the summoner's MP. But still, spirits CAN and have shown to be able to summon themselves without needing the key or the summoner, and by using their own MP.
Why doesn't Lucy ask them to summon themselves? Because they are not strong enough. Leo is an exception, with Virgo staring to do it as well.
That's not even the debate anyway. You're saying Spirits are a magic, that's just false, they are entities, they are people. Sure they live in a different world and are immortal, and can be summoned with a key. But they still have their own MP and their own magic. If they use a magic against August, August can copy it.