Run attached code:

import javafx.application.Application;
import javafx.event.EventHandler;
import javafx.scene.Group;
import javafx.scene.Scene;
import javafx.scene.input.*;
import javafx.scene.paint.Color;
import javafx.scene.text.Text;
import javafx.stage.Stage;

public class dndIssue extends Application {

    @Override public void start(Stage stage) {
        stage.setTitle("Hello Drag And Drop");

        Group root = new Group();
        Scene scene = new Scene(root, 400, 200);
        scene.setFill(Color.LIGHTGREEN);

        final Text target = new Text(300, 100, "DROP HERE");
        target.setScaleX(2.0);
        target.setScaleY(2.0);

        target.setOnDragOver(new EventHandler <DragEvent>() {
            @Override
            public void handle(DragEvent event) {
                /* data is dragged over the target */
                System.out.println("onDragOver");
                
                /* accept it only if it is not dragged from the same node
                 * and if it has a string data */
                if (event.getGestureSource() != target &&
                        event.getDragboard().hasString()) {
                    /* allow for both copying and moving, whatever user chooses */
                    event.acceptTransferModes(TransferMode.COPY_OR_MOVE);
                }
                
                event.consume();
            }
        });

        target.setOnDragEntered(new EventHandler <DragEvent>() {
            @Override
            public void handle(DragEvent event) {
                /* the drag-and-drop gesture entered the target */
                System.out.println("onDragEntered");
                /* show to the user that it is an actual gesture target */
                if (event.getGestureSource() != target &&
                        event.getDragboard().hasString()) {
                    target.setFill(Color.GREEN);
                }
                
                event.consume();
            }
        });

        target.setOnDragExited(new EventHandler <DragEvent>() {
            @Override
            public void handle(DragEvent event) {
                /* mouse moved away, remove the graphical cues */
                target.setFill(Color.BLACK);
                
                event.consume();
            }
        });
        
        target.setOnDragDropped(new EventHandler <DragEvent>() {
            @Override
            public void handle(DragEvent event) {
                /* data dropped */
                System.out.println("onDragDropped");
                /* if there is a string data on dragboard, read it and use it */
                Dragboard db = event.getDragboard();
                boolean success = false;
                if (db.hasString()) {
                    target.setText(db.getString());
                    success = true;
                }
                /* let the source know whether the string was successfully
                 * transferred and used */
                event.setDropCompleted(success);
                
                event.consume();
            }
        });

        root.getChildren().add(target);
        stage.setScene(scene);
        stage.show();
    }

    public static void main(String[] args) {
        Application.launch(args);
    }
}


Open any web-site with image on it. (I used safari)

Try to DnD image from site to the Text Node.

For windows I see output:
onDragOver
....
onDragOver
OnDragDropped.

For mac Isee output:
onDragOver
...
OnDragOver

That's it. Event onDragDropped is not called.