Is in general possible to download data from space missions? I mean "raw" data from sensors instead of the few artistic pictures released to public. Or do scientists keep those files in their secret vaults?
NASA requires all missions to release the raw data on a regular basis, in particular to the Planetary Data System. ESA releases their data to the Planetary Science Archive. Russia releases it's data to the Solar System Data Archive. Other organizations are found under the umbrella "International Planetary Data Alliance", which has a whitepaper here giving the names of all of the organizations with such data.
Certainly it is "possible to download data from space missions".
A great example is the FUNcube program which is sponsored by AMSAT-UK. According to their webpage 'More than 900 stations, including many at schools and colleges around the world, have received and decoded the telemetry.' I'd suggest taking a look at URL https://funcube.org.uk/ which has information about what is involved and links to further details.
And FUNcube is by no means the only satellite you can monitor. http://www.pe0sat.vgnet.nl/decoding/satellite-telemetry/ contains links to a great deal of pertinent "how-to-do-it" information; just choose the "Satellite" dropdown from the menu bar to get an idea of what is possible.