Preheat the oven to 350°F (175°C).
In a medium mixing bowl, combine the softened cream cheese, buffalo wing sauce, and ranch dressing. Whisk until the mixture is smooth and uniform in color.
Fold in the shredded chicken and half of the cheddar and mozzarella cheeses until evenly distributed.
Transfer the mixture into an 8x8 inch baking dish or a 1-quart oven-proof casserole dish, smoothing the top with a spatula.
Sprinkle the remaining cheddar and mozzarella cheese evenly over the top of the dip.
Bake for 20 to 25 minutes, or until the cheese is melted, bubbling around the edges, and starting to turn golden brown.
Remove from the oven and let rest for 5 minutes. Garnish with sliced green onions and blue cheese crumbles.
Serve warm with celery sticks, sturdy tortilla chips, or toasted baguette slices.