There is still a lot of insecureness about this topic.
Unless alot people still think about, the internet is no legal black hole

Private use means, that you are allowed to download artworks and pictures to your own PC. You can then use them on your desktop screen or print them on your homeprinter (though I would not recommend this due to the low quality) You might also use them as avatars on OFFLINE games.. things like that.

Private use does NOT mean that you are allowed to upload those artworks and pictures somehwere on the internet! Not on your blog not on facebook, aswell not on a forum or similar unless you have written permission of the artist

Easiest to say, on that moment you show those artworks in the public, even if you are using those internetpages "privately" you do a copyright infrigment which can end up in you beeing sued by a lawyer.
And believe me, in most countries the artist is well enough protected to take legal actions against copyright infrigment.

Please think up, its not a compliment to the artist if you take his hard work , crop it , change it or use it somehwere for anything on the internet without beeing respectfull enough to ask beforehand. And believe above all it is NOT free advertise for the artist - far far away from this!

If you like to use a painting of mine, then just ask me. I dont bite and in most cases (if its not tube stuff) I am willing to negotiate ;)
Just contact me