THE ANSWER
They use real electrolysis principles, so unlike electric tweezers they are not based on a false mechanism. The difficulty is the technique. Finding the follicle, judging depth and judging current are skills, and most people who try at home do not reach their goal. It is possible. It is not easy, and poor insertion can damage skin.
Why
A home kit typically uses the galvanic method. A probe is inserted into the follicle, current is applied for a period, and the hair is then removed with tweezers if the treatment worked. The underlying principle is sound and it is the same principle a professional uses.
What differs is everything around it. A follicle is not visible from the surface, so insertion means following the hair by feel at the correct angle and depth. Too shallow and the growth cells are untouched, so the hair returns. Too deep or at the wrong angle and you are pressing a probe through follicle wall into tissue, which is how skin gets damaged.
Doing this on yourself adds two more problems. You are working in a mirror, which reverses everything, and you are working one-handed on skin you are also holding taut. Areas you cannot see well are effectively untreatable.
There is also the arithmetic. Galvanic treatment takes considerably longer per hair than professional methods. Users who document their attempts describe hours of work for results that a professional session would cover far more quickly.
What is the best home electrolysis kit?
This page does not recommend a device, because the limiting factor is technique rather than equipment. A better kit does not solve insertion, which is where home attempts actually fail.
Is DIY electrolysis safe?
It can damage skin if insertion is wrong. Both the American Academy of Dermatology and Cleveland Clinic advise having electrolysis done by a trained professional, and that guidance is about the margin for error rather than about the principle.
Can I do electrolysis on my own face?
Some people do, on small accessible areas, with practice. Working in a mirror on your own face is significantly harder than treating someone else, and areas you cannot see clearly are not realistically treatable.
Can I save money doing it myself?
Possibly, on a very small area, if you are prepared to invest a lot of time in learning. Most people find the time cost far higher than expected and the results patchy, which is its own kind of expense.
What the evidence says
Both the American Academy of Dermatology and Cleveland Clinic advise professional treatment rather than self-treatment. Users documenting home attempts on the niche's dedicated forums describe the difficulty of insertion and the time required in detail, including from people who persisted. This page reports that guidance and those accounts, and does not provide technique instruction.
What to do next
If you are considering it, be honest about what you are taking on: a skill that takes practice, applied to skin you cannot easily see, over many hours.
If you have already started at home, the work is not wasted. Follicles you treated successfully are treated. Have the area looked at rather than abandoning it. See electrolysis consultation.