That would basically mean the government funding the satellites, computers, storage, etc. Which is basically where almost all expenditures are. So it would effectively socialize the cost but privatize the results. In essence we would have to pay twice for the data, once to the government in taxes for the equipment, then again to some private company because....why again?
If you have Pay TV, you are already paying twice for the data when it is delivered via your local news station, national news station, or The Weather Channel. Would be nice if NOAA would make some money providing its data to commercial operations.
I can also get it online via noaa.gov or weather.gov, but I still need to have an internet connection. Does that mean I'm paying twice? I don't think paying for access to a telecommunication platform counts as "paying twice" in the same way.
Just to add to this... even if the parent comment's conjecture is correct, under that hypothetical you're already paying twice now, and if they privatize the profits you'd be paying three times.