I may be wrong with some of this info, but this is as much as I know - I'm sure some of the others will be able to chip in too

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What do these qualification letters stand for please, and how qualified is the person.
1 B.D.S.
BDS stands for Bachelor of Dental Surgery and is the degree that all dentists who have trained in the UK have to pass in order to qualify (and practise) as dentists. Different countries have different training requirements and qualifications, but in the UK, I think the training is five years at dental school (university).
Depending on which university the dentist went to, they may have the letters BChD instead of BDS (I know that Leeds uses these letters). BChD is the Latin abbreviation of Baccalaureus Chirurgiae Dentium (which is just a posh way of saying Bachelor of Dental Surgery).
These are qualifications which are gained after someone has qualified as a dentist, meaning that they have undertaken extra training. Dentistry is considered to be a specialism within surgery, hence the title 'Dental Surgeon' (the more formal version of the job title 'Dentist') and also the Faculty of Dental Surgery of the Royal College of Surgeons. Hospital surgeons (for example cardiac or orthopaedic surgeons etc) also undertake similar diploma courses as part of the Royal College of Surgeons, after they qualify as doctors because it enables them to further develop their career in their chosen specialism.
The MJDF
[now MFDS - admin edit 2021] appears to be a relatively new qualification which combines a couple of previous similar qualifications.
The MFDS examines the skills and knowledge required in both primary and secondary care.
www.rcseng.ac.uk
If someone has qualified 4 years ago, how competent are they.
I think this is perhaps one of those how long is a piece of string questions! If someone qualified 4 years ago, then that means they have 4 years practical experience in addition to their university training. They have to have met a certain level of competency and possess a certain level of knowledge in order to qualify in the first place. In other words, they must know what they're doing to a certain extent, otherwise they wouldn't have passed their exams!
I would imagine that dentistry is much like a lot of other careers in that the real learning takes place once you qualify and start to gain experience in the real world. Having spent a few years in the past as a patient at an NHS practice where I saw several dentists (because they never stayed more than a couple of years), I've seen quite a few dentists with varying levels of experience, from newly qualified to around 5 years experience.
In terms of competency (from a patient's gut feeling/perspective about whether or not the dentist knows what they're doing), I would say that in the first couple of years post qualification, they're still quite 'nervous' about getting things right and are not always that confident, because they're still getting used to being in the big wide world of dentistry (even though they probably did work placements during their training). After 2 or 3 years, they seem to settle down a bit and develop a bit more confidence in what they're doing (either that or they develop the ability to give off an air of confidence even when they're not

), whilst still being keen to continue their learning (which is why they often go on to do further training).
I suppose it depends what you mean by competency. Unless they're naturally a good 'people person', interpersonal skills don't always seem to be as honed as their technical skills until they've had a few years experience (I'm speaking from experience here, having encountered the mumbling and not very talkative 'newly qualified dentist' in the past!), but at 3 or 4 years, these things should be starting to come together.
Would someone with these qualifications be welcomed to work in an established private practice, to work on private patients. I need it explained in really dummy laymans terms please.
All dentists have to be qualified and registered with the General Dental Council to work as a dentist in the UK, but I would think that the more qualified and experienced someone is, then that would make them more attractive as a candidate when applying for jobs either in private practice or the NHS... Pretty much the same as in many other occupations.