The EIC (electrical installation certificate) or Minor Works Certificate has to be completed at end of any electrical work but the person over seeing the work i.e. in control so master could include an apprentices work, but he has to be in control so he could not sign for work you have done, and the EICR (electrical installation condition report) is often used to replace the EIC when getting a completion certificate from LABC (local authority building control) but the person doing it is commissioned by them so they can say how rigorous the inspection needs to be. The LABC can issue a completion certificate on the strength of an EICR but there not forced to, and unless an emergency they should be informed (and fees paid, in Wales starts at £100 plus vat for first £2000 worth of work) before the work starts. Any one can issue an EIC, EICR, or Minor Works, but if a rental property in England then there are extra requirements for the EICR to be able to be used to satisfy the new law. But in theory there is nothing stopping anyone doing any electrical work which they are capable of doing on a domestic property and getting the required paperwork, even if not really an electrician, the only problem is it is uneconomical to pay the LABC fees for every job.
So an electrician does not need to be a scheme member, and it clearly costs to be a scheme member, so if doing mainly commercial or industrial it would not be worth the annual fees to be a scheme member, the annual fee is not cheap, and although in Wales it would be hard to do much electrical work on domestic without being a scheme member, in England there is a lot of work that does not require registering so it may simply not be worth a sole trader paying the annual fees, a good electrician can likely find enough commercial and industrial work not to need to do domestic. Big difference is commercial and industrial is a lot of repeat work, so unless good you would run out of work quite fast, where with domestic you often only work for some one once, and there are many more unsuspecting customers so a bad electrician could without the Part P work for years without his poor reputation catching up with him, so the schemes are designed to ensure poor electricians can't practice for long. Not that it works, but that was the idea.
So a scheme member can get a compliance certificate issued far cheaper than a non scheme member can get a completion certificate issued, the problem is it is not up to the electrician to inform the LABC that is up to the owner, same applies for building work, so if he does a safe job, and issues a EIC or Minor works at end of job, he has not broken any laws and long as he does not claim to be a scheme member, there is no permit to works system so if you tell him you have registered the work, he just takes your word for it, so complaining he has not had a compliance or completion certificate issued get you no where, as all he has to say is he thought you had registered the work, and unless he admits he knew the work had not been registered there is very little you can do. However since he knows you would need the EIC if he does not issue one, then it is really an admission of guilt. But BS 7671 is not law, so there is nothing to force him to issue one, except by not issuing one he is admitting he knew the work had not been registered.
There is one extra quirk to this, to be a scheme member you have to agree to follow BS 7671 so for scheme members BS 7671 is law. So if I wanted to wire a house to German standards I would have to do it through LABC as the schemes would not permit it.
So returning to the EICR this is some ones personal option as to the condition of the installation, unlike your MOT he does not have a rule book he must follow, there are loads of guides, but unless a scheme member, he does not need to follow them, for a rented domestic property in England he must have insurance and the government report states they expect at least 10 years in the Electrical trade, but this has not been written into the law, there was going to be a government check list that he had to satisfy, but I have failed to find one, I can find loads issued by private bodies, but not a government one. But unless rented anyone can complete an EICR, and there was a report on the electrical condition when I got my house buyer report, although rather poor.
Within the trade the C&G 2391 was considered to show a person was qualified to make the report, but what matters is that some insurer is willing to issue professional indemnity insurance to cover the guy doing the report. If you look at any forum there are arguments as to what should get a C2 and C3 galore, and for a non rental it does not really matter, all you want to know is what work needs doing, and what work should be considered needed if extras are added to the installation. If not rental then it does not matter how many C2's you can ignore them if you want.
As to if worth having a formal EICR doing not so sure, any good electrician will test anyway before starting to fit a consumer unit, last thing he wants is at end of the job, the RCD will not hold in and you refusing to pay him until it does, so he wants to know problems before he starts, fitting an all RCBO consumer unit then only faulty circuits would not work, so he may get away with that, but with a common RCD for many circuits a neutral - earth fault on one circuit could stop all good circuits being used. So a good electrician will do a number of tests before starting, including looking for borrowed neutrals, it is not as good as a full EICR but last thing you want is to pay twice, so if you want the report best would be to get the electrician to do both fitting of consumer unit and writing out an EICR as well as the EIC at end of job, and of course a completion or compliance certificate, and to issue them at a reasonable cost he would need to be a scheme member. And you want any EICR to be done on scheme members paperwork, so should you sell the house, the buyer knows the person doing the inspections was both qualified and insured to do the work. And because controlled by the scheme provider he should be following their guide lines as to what has a C1, C2, C3 or F1 coding.