Applescript code to find path to Desktop picture in Desktop 2
This works for me:

tell application "System Events"
set tlst to a reference to every desktop
--set picture of item 1 of tlst to newtop  -- display 1
--set picture of item 2 of tlst to newtop -- display 2
set z to (picture of item 2 of tlst) as text
end tell
return z


The commented out lines are for setting desktop pic on one display or the other. newtop is obviously preset, or the result of

set newtop to choose file