Technically speaking: yes, posting pictures from Pokémon episodes would be illegal. Shogakukan could send a cease and desist to let's say PM because they do not own the rights of the Pokémon anime.
The same would go if you right-click on an image at Pokemon.com and save it to you hard disk, because you do not have the right to copy the image, thus copyright infrigment had been taken place.
In reality, I do not expect it would go that far. The witch hunting for illegal music and video is because people wouldn't buy the movie/music if people could download it from the Internet (wether this is true of false is an other discussion).
It's not like Shogakukan is selling pictures of the Pokemon anime. So, Shogakukan wouldn't gain anything by starting a witch hunt for sites that post episode images. If it should happen, larger sites like 2chan or Random Curiousity would disappear first, since they cover episode images from basically every anime. On that level Serebii, PM, PokéNightmare and Filb.de are small fish in a big pond.
Basically the sites are leaning on the goodwill of the anime right holders.
However I can imagine that action would be taken against full manga scans, because Viz Media is selling manga's.
On a side note: the webmaster(s) probably meant this for their own created content like news reports, walkthroughs, episode summaries etc but I find it rather amusing to read: "All Content is ©Copyright of Serebii.net 1999-2008." at the bottom of a page filled with episode screencaps. XD