The probes sent by NASA generate a lot of data for scientists to study, but how available is it to non-NASA affiliated scientists?
For instance, would I (a layperson) be able to get the raw data collected by the Mars rovers?
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According to the NASA webpage Open Government Initiative, there are several repositories of raw data from NASA missions. These are sorted by categories, however, as this site is a bit dated, the links are sometimes broken - but is included to give you an idea what is actually there.
The official US government website Data.gov also has over 3000 data sets from NASA alone.
As a specific example, about the Mars rovers, an example of the data available is from NASA's PDS: The Planetary Atmospheres Data Node (Scroll down the page and you'll see the Mars rover data for this context.
All NASA data is to be released to the public within 1 year of it's collection, and is released to the Planetary Data System. The 1 year is to give the scientists a chance to publish on the research they have worked long and hard to colelct.
The Mikulski Archive for Space Telescopes (MAST) contains a considerable amount of raw and normalized astronomy data including that from the Hubble Space Telescope, GALEX, FUSE, and Kepler.