The easiest way to show overlapping points is to set layer rendering style to
Point Cluster. You will automatically see how many points are clustered in one location.
This works for points on the same layer. So first create a new layer, combining all point layers, using Menu Processing > Toolbox > Merge Vector layers with all point layers as input. Then apply Point cluster style to this layer to see where points are overlapping.
Instead of this, you could also create a virtual layer to create a layer containing all points - similar to the solution by @Bera, but with an easier query, without counting the points. This part is done by the point cluster renderer:
select points1.geometry as geom from points1
select points2.geometry as geom from points2
select points3.geometry as geom from points3
Blue dots: single points; red dots: showing number of points from all three initial point layers, clustered in this place: