I am creating a screen with four lists on it. Basically two pairs of lists where you can select lines on one list in the pair and move them to the other list in the pair.
Looking at the documentation I need a ListSelectionModel for each list to determine which lines have been selected. I will use a [Sel] or [Des] button to do the actual transfer.

The documentations and samples say I need a ListSelectionListener but as I will not access the model until the user clicks on the button do I actually need a listener? Will the model still have the getMinSelectionIndex, getMaxSelectionIndex and isSelectedIndex set if I do not have a listener?
You do not need a listener, a listener is only useful for keeping something in sync elsewhere, which you don't need.
You can access the selected indexes at any point after the selection event(s) occurs. The method JList.getSelectedIndices returns an array of currently selected indexes, and getSelectedValuesList() returns the actual items depending on what you want....
JList<String> items = new JList<String>(new String[] { "foo", "bar", "baz" });
// simulate selection
items.setSelectedIndices(new int[] { 0, 2 });
Sometime later....
// get actual values
System.out.println(items.getSelectedValuesList());
// get indexes
System.out.println(Arrays.asList(items.getSelectedIndices()));
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