Most people would say that they jumped off the swing but why not jumped from the swing?
If someone jumped off the swing, that means that he was previously on the swing. Jumping from the swing could mean that he was next to the swing. I could stand next to someone and say "roar". If I succeeded in startling him, he might jump from me. If he is standing next to the swing as someone riding the swing starts coming down, he might jump from the swing to avoid getting hit. Jumping from the swing isn't used as much because it is ambiguous.