It's quite true that the various rules are inconsistent. Ridiculously so, in fact.
More than likely, your product needs to be CE marked. This mark indicates that you declare the product to be compliant with all applicable EU directives.
In terms of electrical safety, this will be the Low Voltage Directive, and the applicable standard is EN 60950 (or some other related standard, depending on the specific type of equipment you're producing).
EN 60950 (or whatever) defines the technical characteristics your product is required to meet. It specifies things like isolation and earthing requirements, flammability, creepage and clearance distances, and so on. You should ensure your product is designed according to this standard. If you desire formal documentation that your product meets the standard, then you may use a 3rd party test lab to test it for you - but the responsibility is still yours even if you have a report that says "compliant" on it. In the event that your product hurts somebody, that report is a good defence against a claim of negligence or non-compliance, but the buck still stops with you.
Since you are the one applying the CE mark, you are responsible for declaring that the finished product is compliant. There is no specific certification that the person assembling your equipment needs to meet - at least, not in a legal sense, and not for most ordinary commercial, industrial and domestic equipment.
You, however, may wish to impose training and/or certification requirements on your assembly staff, if that helps make you confident that they are able to carry out the work safely and correctly. The reason is simple: you're attaching the CE mark, so it's your neck on the block if they screw up.
(Note: by "You" I'm referring to the person whose name appears on the CE declaration of conformity, which is a certificate you should include with the product or in the manual. Typically this would be the Technical Director, CEO or other senior member of staff who has the authority to stop production if necessary. Responsibility for compliance is personal... in the worst case, the person whose name appears on that certificate goes to jail).
And yes, it's quite absurd that I can't freely work on the electrical wiring in my own home, but I can build mains powered products and sell them to you without having done any testing on them whatsoever - provided I attach a CE mark and thereby declare them "safe".