How portraits appear
J.K.Rowling explained that portraits normally appeared in a magical way:
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.
--J K Rowling at the Edinburgh Book Festival, 15 August 2004.
Her later explanation about the Snape's portrait was not so detailed:
Laura Trego: Was the absence of snapes portrait in the headmasters office in the last scene innocent or deliberate
J.K. Rowling: It was deliberate. Snape had effectively abandoned his post before dying, so he had not merited inclusion in these august circles.
J.K. Rowling: However, I like to think that Harry would be instrumental in ensuring that Snape's portrait would appear there in due course.
--J.K. Rowling and the Live Chat, Bloomsbury.com, July 30, 2007 (2.00-3.00pm BST).
My theory is that a powerful wizard could have probably reinforced the imprints that Snape left behind, attaching them to a hand-drawn portrait or even magically creating the drawing. They could also create and animate the portrait artificially, using the same magical technique as Professor McGonagall in the first book (when she tranfigured magical chess figures to make them alive).
At least in two cases, the portraits appeared in two places:
- Professor Phineas Nigellus Black: Hogwarts Headmaster's office and Grimmauld Place.
- Professor Dilys Derwent: Hogwarts Headmaster's office and St Mungo's hospital.
We can note that all of these buildings are magically powerful and both persons were strongly linked to these places.
Other instances of their portraits might exist somewhere, but we don't know this for sure. Professor Dilys Derwent could also have an ancient mansion with her portrait in it, who knows.
I am pretty sure that all of the portraits appear 'automatically' and are magically linked without any need to perform an incantation manually.
How it works
There might be two or more portraits, but the person is unique. When it moves to another portrait, it disappears from the current one.
Obviously, the person would look the same if you catch him or her on any of the portraits.
Apart from ability to move between their own portraits, they can move to the neighbouring portraits, at least at Hogwarts & at the Ministry of Magic. Right after the incident with Arthur Weasley:
But Dumbledore stood up, so quickly it made Harry jump, and addressed one of the old portraits hanging very near the ceiling. 'Everard?' he said sharply. 'And you too, Dilys!'
[...] 'Everard, you will need to raise the alarm, make sure he is found by the right people -'
Both nodded and moved sideways out of their frames, but instead of emerging in neighbouring pictures (as usually happened at Hogwarts) neither reappeared.
And at the Ministry:
He doesn’t look good, he’s covered in blood, I ran along to Elfrida Cragg’s portrait to get a good view as they left.
--Harry Potter and the Order of Phoenix, chapter 22