As for the DRF, there is no doubt in my mind: It's word of mouth or otherwise through an extended network. And that goes all the way down to the cleaners.
When we are talking about senior court officials, it's people who are already known to some degree by other advisers or court officials. They are then approached to see if they are interested. That's how the current Chief of Court was employed.
Other senior court officials have a military background, requiring talent for organization and leading people. That's how the Masters of Ceremonies have been employed.
As for footmen and employees at the royal stables. Some of them are brought in when young. In at least once case after having served as a volunteer conscript with the Guards Hussars. The women in question then served at the stables before becoming a footman.
Then there are cleaners. They are almost invariably people who have worked for the state, say in ministries before and as such are already cleared and who potentially could have had access to confidential info. Working for the court is a relatively cozy job as a cleaner. You don't have to run ever faster, but can take more time in doing the job properly - a luxury few cleaners have nowadays. And often an employee at the court have access to rent an apartment at the Amalienborg complex at a very affordable price, considering the location!
Such a post formally has to be advertised, but in reality those who seek such a job is rarely an outsider.
In short: It's people who are known, recommended and approached and encouraged to seek a specific position at court.
Naturally all employees have to cleared by PET, that's no secret and as a state employee they have to sign a letter of confidence, making it a criminal offense to talk about sensitive matters to anyone.
M&F's cook for example is the husband of a friend of M&F.
Then there are the staff of adjutants. Being an adjutant for the DRF is not a temporary posting as in other countries. They apply and if accepted basically can remain adjutants until retirement or they seek another post. So a number of QMII's staff of adjutants have been around for years.
Just as with being the Captain of Dannebrog, it's usually the last position they have before retirement.
The drivers however are seconded from the military. I have an army-buddy who now drives or the DRF. They are usually recruited among career NCO's who have served for a long time and who now has this as a little cozy job, before retiring or returning to their unit. It's considered a little reward - it also means that you (and your family) don't have to relocate to a new posting at least for a few years.
These drivers undergo special training in evasive driving at FET (The military intelligence service) and a course in protocol.
Then there are the Ladies in Waiting. They are personal friends of the female royal in question. As it is considered an honor to serve the royal family they are not paid a salary, but only get expenses covered. They are not employees, they are above that.
As for nannies, and here speaking of the nannies employed by M&F, that must clearly have been through a network. Mette Hansen, is by profession a very qualified nurse at Rigshospitalet. So someone would have known her and recommended her.
At least one nanny, a British nanny, was employed through a very renowned British agency. They are highly trained, highly recommended and very discreet.
Mary's stylist and certainly former Chamber Maid, was recommended through the network of her hairdresser, Søren Hedegaard. The now dead Chamber Maid came from the world of theater and Søren Heedegaard's husband knew her very well.