I have seen it done both ways:
- The background content is blurred, and the pop-up is semi-translucent.
- The background content remains clear (no overlay) and the pop-up itself has a blur.
For the left image, with a blurred background, the focus is drawn to the pop-up, and I could be wrong, but it feels that this more accurately mirrors what our eyes do, focusing in and out of foreground and background.
For the right image, this is sort of going in more the "frosted glass" direction, perhaps more a of a modern / sci-fi feel to it as well. I should add that the left image needs a bit of work with shading etc so it pops a bit more.
Is this down to personal designer preference, or is there some solid thinking behind choosing one over the other? This would be for software use.