I can not answer for all of Europe because the process may have varied but I can answer for Sweden. The executioner was usually recruited the same way any other employee of the city was, there was no mystery involved. Usually the executioner himself was recruited from among the helpers of the previous executioner.

Executioners had some manner of responsibility, under him he would have helpers called "rackare" and "rackardrängar" (or "Nattmän" ('Night men' because they took human refuse out of the city at night)) in descending order of importance. These were men who would handle corporal punishment in various forms and the clean up. These men also handled burials of suicide victims, killed horses and often handled the city waste. The executioner himself often handled only the executions themselves and oversaw the work of the his 'employees'.

It was however a low status profession and recruitment was periodically difficult. In the 17th century in particular it was very difficult and the practice of postponing the execution of a prisoner on death row in exchange for service as executioner was introduced. If a prisoner accepted, his ears were cut off and he was branded with the seal of the city. However, the prisoner was not actually pardoned, the execution was merely postponed. Oftentimes the first act of the new exectioner was to execute the previous one. Executioners who served diligently and faithfully were occasionally pardoned at the end of their service but this was by no means a guarantee. This practice was however outlawed in 1699.

I hope this answers your question at least in part. I'm not at home at the moment and wrote this up from memory, if you want sources in Swedish, let me know and I can provide them in a few hours.

Maybe not 100% what you asked for, but here goes:

Being an executioner in medieval/early modern history was not a glamouros job. Their connection and everyday interaction with death made them social pariahs, so much in fact that in many cities and villages, executioners had their own seperate table at the local in. But how did one become an executioner?

Like many things, your personal career (provided you were a peasant) was decided by the estate/ the family you were born in. A merchant's son would most likely become a merchant, a butcher would follow his father's footsteps and become a butcher himself.

Executioners were no exception to this rule, so son's to executioners would most likely be raised to become executioners themselves. This lead to "Executioner families" with generation after generation picking up the trade. The Scholl siblings (german anti-nazi resistance) were actually executed by one such member of a century-old executioner family.

To summarize: Being an executioner was not a glamouros job, and you would most likely be born into it.

Source: Van Dülmen, Richard (1995) "Theater des Schreckens". Becksche Reihe