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It depends how the image is made. It sounds like you already know that they are 2 rectangles side by side. But just by looking at your image, it could be a larger rectangle with a line down the middle. So assuming they were originally 2 rectangles with the shared side on top of each other; and assuming it's been changed to path with Object to Path, here's what you can do. Note that there are probably several ways to do it, and this is my choice.
I would select one rectangle, click on the side that I want to remove, and click "Delete segment between 2 non-endpoint nodes" button. Then do that for the other rectangle. Now if you want the pieces to be connected, so that it's one piece, I would join the nodes. Using the Node tool, select both pieces (hold Shift for multiple selection). The draw tiny selection box around the 2 nodes that you want to join, then "Join selected nodes" button. Do that once for the top and once for the bottom. And if you want a true rectangle with 4 nodes, you'll need to delete the node created by joining 2 nodes.
If the original 2 rectangles are grouped, there will be a certain way to select one at a time. If the original 2 are subpaths, you won't need to make a multiple selection in the 2nd step (joining nodes). Instead of joining nodes, some people would use Union. However, if the original 2 rectangles were not precisely sharing their sides, Union might result in not a true rectangle. But if you happen to know that they precisely share their sides, Union will be faster than node editing. Or if you don't need precision, Union will be faster.
Actually....well if precision isn't needed, or you know that they were true rectangles originally, you could select the original 2 rectangles, and do Path menu > Union, right from the start!