The Duplicate Geometry Check will consider your two point features to be duplicate geometry when they are colocated, or are at the same location (i.e. have the same X,Y). If they have different X, Y, the Duplicate Geometry Check will not detect your two point features.
If you want to find point features that are near each other, have you considered running a the Geometry on Geometry Check with Spatial Relation Check set to Intersects within a specified Tolerance?