At my daughter's house at this time finishing up a painting job. I decided to test the wall plugs for correct wiring and here is my question.

Two plugs, opposite walls. One of the outlets on each is normal using the plug in tester, one tests and indicates open ground but the test light on the plug lights up dim.

So what do I look for? Presently at her house and trying to figure out what to look at next.

If it were me, I would turn the breaker off that feeds this outlet...checking to make sure both outlets are powerless. Then pull the outlet out from the box and see what the difference is.
It is possible that the bottom outlet is switched, but that should not affect the ground.
You might be able to just look at it and see the difference.