There's no difference between trading rights, and trading the player. You trade one, you automatically trade the other.
The reason Malkin can't be signed by another team, as has been mentioned, is because he's not a free agent.
Well no. There is a pretty substantial difference. If you trade for the rights to Evgeni Malkin, you have the sole rights to negotiate an NHL deal with him, but you don't have him. He can go sign in Russia or Sweden or Swizterland or wherever he wants. If you trade fo Malkin himself, that means he is already under an NHL contract, and will therefore be bound by contract to your team.