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
union all
select points2.geometry as geom from points2
union all
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: