You need a node path in order to access the scene. It works the same as accessing objects.
So something like this near the top of the script on the scene 1 button:
onready var my_sprite = get_node("/root/Game/Scene2/Sprite")
var new_texture = preload("res://new_texture.jpg")
On this pressed function of the button:
my_sprite.texture = new_texture
Hope that helps.