Mom sent home from hospital with o2 for use when she is active/walking. Now re tests show she doesn't need the o2. How can I be sure now?

Depends... There are situations in which oxygen levels can improve over time. For example, if your mom was recovering from pneumonia, acute bronchitis or bronchospasm from either asthma or copd, her oxygen level may have been low enough to warrant supplemental oxygen but as these conditions improve, the oxygen level improves as well. She can be evaluated by her doctor to be sure she no longer requires it.
02 saturation. The test that is done on your mom's finger with the little red light (the 02 pulse oximeter) should be a good guide of whether she needs the extra 02 or not. Generally, if when she's active/walking and the 02 stays above 93%, that's pretty good, and insurance usually wouldn't pay for extra 02 at that point anyway.