If I may add something to that response;
go through the mind numbing B.S. that is required to get a flight training operating certificate?
How hard is it to fill in a simple form with the details of instructor(s), aircraft and insurance and wait for the free certificate?
I feel that it is only right that the Authority responsible for oversight has atleast a note of who is doing what in the area of training.
Don't forget that the local personal supervision of every pilot that you may be familiar with on the west side of the Atlantic is not done in Europe. i.e. there is not an official person located on (or very close to) every reasonably sized airport watching what you do.
These is also a different culture (in the UK especially) with regard to flying. By this I mean that while in the US if a PPL is seen by a CFI to be doing something stupid, the CFI will have a word with said PPL. In the UK, there is lamost zero chance of that happening with an FI and PPL.
In other words, in the US, the CFI will take action because they kow if they don't the FAA will take action and ask why the CFI did not. Hence lots of people very worried about certificates being suspended, the feds catching them if they do something wrong etc. In Europe the FI's will ignore most of what goes on and the CAA are not there to see it anyway so there is a greather posibility for stupid things killing people. Ergo the common statement on sites such as this (Private Flying Forum) that "it may be breaking the rules/ the law / not 100% safe but who is ever going to see you do it".
Sending the form off at least gives the Authority the ability to pretend to be surprised if they turn up at a muddy field and find a non-licence holder training another non-licence holder.