Where and how were the catapults/trebuchets operators trained?

It is a truism that the source of most training was from master to apprentice, who then went on to become a master, taking on an apprentice – and so on. But, the question still remains: How did the masters train their apprentice?

But the siege artillery is different: It is huge, heavy and expensive to operate - you can't order your average Jonny The Peasant to practice shooting from a trebuchet every weekend to get some practice in case of war... Maybe there were guilds for such engineers? Or maybe it was a knowledge passed from father to son in a selected few families?

In his book, The Medieval Military Engineer, Purton espouses that:

... it remains most likely that a large majority of master craftsmen acquired their skill through apprenticeship to an existing master. Some of these crafts began to specialise and then adopt the structure of the guild in order to protect their interests – although from the viewpoint of skills with military application, it should be noted that miners and masons remained outside these structures. The engineers who will have begun as master craftsmen themselves may have learnt their trade at the feet of existing engineers. By learning on the job during military operations – there would have been no shortage of opportunities – perhaps when they became masons or carpenters they engaged in the construction of engines, observing close up and sufficiently sharp-witted and intelligent to remember what they saw, and sometimes to suggest ways of making the engine more effective, sometimes to help in the diffusion of knowledge and experience from one region to another.

The master would've also told his apprentice the fundamental principles of which was pertinent to the things he would've done. The master builders’ and carpenters’ knowledge can only be guessed at since not all were literate: builders must have had at least a little mathematical knowledge and been able to interpret the drawings that almost certainly must have been provided for them by the patron.

Collections of the illustrations of building construction from medieval manuscripts demonstrate that building workers (covering by this term everyone from the master builder to the newest apprentice and simple labourer) were acquainted not just with appropriate tools, but also with equipment to allow them to use their skills. Hammers, saws, trowels, axes – all are repeatedly shown such as this one:

Therefore, if Jonny the Peasant needed help, then Jonny the Peasant could ask his crew for help.

But how was this education achieved? While European universities existed as early as 1088, I'm not sure that they were not granting titles of "combat engineer".

During the Middle Ages, Christendom permeated the notion that knowledge of the subjects of the quadrivium (arithmetic, geometry, music and astronomy) was an essential part of school-learning designed to glorify God and to educate clerics, and that it was purely theoretical: scientia had nothing whatever to do with the skills practiced by common craftsmen and artisans – the artes mechanicae. Thus, the literate people perceived the mechanical arts as stated below although there were a few exceptions:

... the mechanical arts necessarily were the responsibility of ignorant commoners, and the people who practised them were merely tolerated because their labour – drudgery, of low status, mindless: something vulgar – was necessary to society. Many of the attempts to classify the sciences since the days of the Roman empire maintained this distinction, originating in the approach of the Greek philosophers, and excluded the mechanical arts altogether from their lists. But some did not, and mechanical arts were included not just in the lists but sometimes in the practical application of the teaching of the “liberal arts".

The exception being Hugh of St Victor who advocated for the recognition of the importance of the mechanical arts and described their application. Many other medieval scientists made contributions such as Adelard of Bath et al.

I suggest you read The Medieval Military Engineer by Peter Purton since it has more fruits to share which can plausibly quench any other questions you may have.

Source: The Medieval Military Engineer by Peter Purton