According to rems answer to this question, J.K. Rowling already answered.
Q: All the paintings we have seen at Hogwarts are of dead people. They seem to be living through their portraits. How is this so? If there was a painting of Harry’s parents, would he be able to obtain advice from them?
JKR: That is a very good question. They are all of dead people; they are not as fully realised as ghosts, as you have probably noticed. The place where you see them really talk is in Dumbledore’s office, primarily; the idea is that the previous headmasters and headmistresses leave behind a faint imprint of themselves. They leave their aura, almost, in the office and they can give some counsel to the present occupant, but it is not like being a ghost. They repeat catchphrases, almost. The portrait of Sirius’ mother is not a very 3D personality; she is not very fully realised. She repeats catchphrases that she had when she was alive. If Harry had a portrait of his parents it would not help him a great deal. If he could meet them as ghosts, that would be a much more meaningful interaction, but as Nick explained at the end of Phoenix—I am straying into dangerous territory, but I think you probably know what he explained—there are some people who would not come back as ghosts because they are unafraid, or less afraid, of death.
source from Accio quote, Edinburgh Book Festival, 2004
To sum up, let's say that paintings are like memories. People on paintings can only "repeat catchphrases", because they leave a part of their aura behind them, and only in places that were highly related to them. So yes, in a way, they leave something that could be assimilated to "a part of their souls", but that's all. They can give some advice, but nothing that you can't remember or expect from them.