There are several reasons to have black spots and all are related to the removal of the hair root.
When hair was removed with the root in the growth phase -anagen-(with tweezers, wax, thread, …) one of the sheaths surrounding the hair root (outer sheath) remains on the path, ie,it is trapped inside the follicle.
When the hair is dark, the sheat contains a bit of pigment too. The following hair to be formed with a new cycle will push the sheat to the surface of the skin. And you can see a black spot on the opening of the follicle. If you scratch or drain a little, you will see that the little dot is released to make way for new hair tip.
You can also see this in a treated area with Electrolysis, the difference is that the black point can not be pushed by other hair, but the scar tissue to repair damaged tissue in the follicle and eject the pod to the outside. This does not happen when the hair is being treated in the early stages, because the sheath is strongly attached to the hair root. This occurs when the hair is being treated in the later stages of growth because the sheath does not have much grip.
Conclusion: If you check that there are black spots on a new hair, change your electrologist, because treatment is not working.