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This is a Photoshop question.
I created a rectangle in a layer. Then I copied it and moved it until it touched the first one along an edge. I would like to be able to stroke each rectangle individually. However, when I select one path and then stroke it, the stroke goes around the outside of the two rectangles together, and the edge along which they touch is not stroked.
How can I stroke each independently?
From my understanding of your question you want to create to identical rectangles, both with a stroke. It sounds like you could create a rectangle with a stroke and duplicate that.
Edit Convert to Smart Object and Rasterize (Right click on the layer in the layers panel to get these options) the layers before you bump them together. I am not having this problem at the minute but I have had similar in the past and I'm sure that this is how I overcame it.
I just tried to recreate your rectangles as I understood them from your description. I can't reproduce the issue you are having, but here are some tips.
Make sure you only select one path at a time. If your rectangles are both in the same path 'layer', the stroke option will say "Stroke Subpath" rather than "Stroke Path" when you have only one rectangle selected. It might make it easier if you keep each rectangle in a separate path 'layer'. Once you have them positioned, you can copy/paste them between layers and they won't move.
It sounds like you may have your path selection tool options 
set to "add", since the shared line is being removed. This would only happen if you have multiple rectangles selected, though.
