Polyps discovered in Philadelphia were initially described as a separate species, but were later determined to be a form of C. sowerbyi (Boulenger and Flower, 1928). So there is only 1 species that I am aware of in the United States, polyps and resting bodies are probably translocated accidentally with stocked fish and aquatic plants or by waterfowl.
Craspedacusta sowerbyi was discovered in the Huron River near Ann Arbor, MI, in 1933, and in Lake Erie shortly thereafter. These hydrozoans have been here a really long time, and apparently they most likely got here by natural means in the 1800s or earlier, probably by fish, birds, or plants or a combination of all three. They don't fit the actual definition of an invasive species however because they are common outside their native range globally, it is rarely in harmful densities to a native habitat, which invasive is defined as being in "harmful densities".
I am of the belief that they most likely originated in the Yangtze River but made there way here naturally many, many years ago. To be a native species its presence in a region must be the result of only natural processes, with no human intervention. And they are not endemic to a specific region so I would consider them to be a naturally occuring, native species.