I can only speak about the Spanish Inquisition, maybe in other countries things were different.

First things first, as the Inquisition did not have their own torturers or executioners, so for the tortures they had to use the Civil Justice executioner, who was also the person that administered the torments. As for the executions, the Inquisition could pronounce death sentences, but could not carry out executions, so the people sentenced to death were given to the Civil Justice so the civil authorities would execute them. This sentence was called "relajar al brazo secular" id est "to be relaxed to the secular branch".

The public executioner was a civil servant, employed by the Justice system, be it paid straight from the Crown's budget or from the Court's budgets, depending on the gig in question. This job had a very bad reputation, so bad in fact that it was relatively common to be in shortage and executioners had to be brought from outside the village. The Crown even tried to fight these shortages of executioners. In 1435, King John II of Castile declared the executioners to be exempted from taxes in order to try to have someone apply for those positions. In 1500, the Catholic Monarchs established that every court should have a crier and an executioner, and that the executioner receive a fair compensation. In ordinances from 1525, and ratified in 1566, some bonuses were approved: the executioners were entitled to the clothes that the executed person was wearing at the moment of death. A price list was established, listing how much would the executioner be paid for each action, but it was normally along the lines of 1 real per person punished, plus the bonuses.

Even with all of this, the position was of low social standing, and very much of shame. So shameful was it, that executioners would have to carry a little stick with them when going to the market, so they would not touch the produce. This low social standing, and social marginalisation produced the obvious effect: it became a marginal job, and more often than not, it would pass down from father to son. It was not very sofisticated, so it could be easily learned. The Spanish Inquisition only had three methods of torture: the strapado, the rack, and waterboarding. The strapado is a very simple pulley, the rack I shall not explain for it is extremely famous, and waterboarding was made famous by the CIA. As for the corporal punishments, non lethal, the Inquisition only administered whiplashes, although they were normally not administered with a whip but with a stick. This was also frequent in the Civil Justice, as made very clear by literary works such as the Lazarillo de Tormes, where Lázaro's stepfather is given "el acostumbrado centenario" ("the usual centenary", one hundred lashes) for stealing.

I did not want to make this answer excedingly long, so I went for clear points.

Sources:

Savater, Fernando and Martínez-Fresneda, Gonzalo (1982), Teoría y presencia de la tortura en España. Barcelona: Anagrama.

Pulido Fernández, Ángel (1897), La pena capital en España. Madrid: Establecimiento Tipográfico de Enrique Teodoro.

Eslava Galán, Juan (1993), Verdugos y torturadores. Madrid: Temas de Hoy.

Deleito y Piñuela (1989), La mala vida en la España de Felipe IV. Madrid: Alianza Editorial.

Arenal, Concepción (1983), El reo, el pueblo y el verdugo. Santiago de Compostela: Biblioteca de Autores Gallegos.

VV.AA (1667), Compilación de las Instrucciones del Oficio de la Santa Inquisición. Madrid: Diego Díaz de la Carrera. Digitised here.

Anonymous (1554) La vida de Lazarillo de Tormes y de sus fortunas y adversidades. Medina del Campo: Mateo y Francisco del Canto.