If there are 2 points on top of each other in QGIS, only one of them will be labeled
How to make it so that both (or more) points show a label?
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Have you ever tried to use the rule-based labelling? Doing so, you will be able to define a different position for each label. So you have to build the proper rule for every entity/point.
To do that, once you are on the Layer Properties > Label, you have to click the green cross button at the bottom of the window, give a name to the rule (Description), define the appropriate filter ("name" = 'point 1'), and then click on Placement tab, option Offset from centroid and adjusting the parameters in order to place your label (quadrant + Offset XY). Once done, click and Ok and you should repeat the process defining a new rule for the second point.