If memory serves (which it may not) there are several instances in the Harry Potter canon of characters conversing with portraits in a manner that implies independent sentience. The Fat Lady outside the Gryffindor common room is one, and the past headmaster portraits in the headmasters office is another.
What I'm curious about is whether there's any evidence as regards whether these portraits were actively intelligent and self aware or merely very clever "recordings" of the subject that mimicked how they were in life?
If the former is correct, it implies that wizards captured in portraits may have been aware of their own death, and have achieved a peculiar form of immortality. In which case you might expect the majority of wizards to have one painted, and perhaps for there to be rather less mourning in the event of a wizard with such a portrait passing away.
It also raises the question of how and why these portraits were different from the photographs in the wizard world. Both could animate their subject matter, but there's never any suggestion that anyone could converse with the subjects of the photos like they could with those of the portraits. Why the difference?