So you have two 2-wire Romex in each box using only three of the four conductors. Usually, there is a 3-wire between the two boxes, but it is not unusual to see two 2-wires run instead.
You believe that those three conductors are connected to each other box to box. While they might be connected electrically, they can’t be connected directly physically, as that just doesn’t make any sense. Basically, you would have this:
So those conductors may not be conducted as you think, because obviously you can’t just connect two switches together like that. They could still be going through another device and be electrically connected so when you continuity test them, you get the result you did.
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Are you sure this is a 3-way? i.e only two switches. Could this possibly be a 4-way with a switch you are not aware of? (I know, sounds ridiculous, but we’ve had people find switches they didn’t know about in back of beds, etc.)
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It’s weird that you can’t find a single hot in either box. If you are using a meter as opposed to a proximity tester, test between the black screws and the stub of the ground, if you haven’t already. Not sure how you tested previously.
@harjms ???