Is there a way to use nodes to combine two images if both the images have transparency (i.e. alpha channels)?
You can use an Alpha over node to automatically handle the alpha mixing:
However, if all you want to do is mix the alpha channels, a normal mix node will do (think of the alpha channel as a gray scale image):
There are a few ways to do this. I will show you the way that worked for me. I drew some images in photoshop with alpha. Here is how to combine two images with alpha using the blender compositor. I found that this is the most reliable method to use for external images.
Final Node Setup: