I'll quote myself from this old post: How were engineers trained in the Roman Republic/Roman Empire, and how much do we know about their design practices?

Vitruvius begins his book On Architecture about the education of the architect. He points out that they need both practical, manual training as well as deep study of theory.

He says that an architect needs to be trained in drawing, geometry, history, philosophy, music, medicine, law, astronomy and arithmetic. Some of these might seem strange for an engineer / architect, but Vitruvius gives examples why they are important from a general standpoint:

As for philosophy, it makes an architect high-minded and not self-assuming, but rather renders him courteous, just, and honest without avariciousness. This is very important, for no work can be rightly done without honesty and incorruptibility. (I.7)

As well as for more specific architectural reasons:

The architect should also have a knowledge of the study of medicine on account of the questions of climates, air, the healthiness and unhealthiness of sites, and the use of different waters. For without these considerations, the healthiness of a dwelling cannot be assured. (I.19)

Vitruvius hints that the training involved many teachers. The practical labour probably occurred as an apprentice under a master engineer. Yet the other disciplines, like history and philosophy, were probably taught by expert tutors in those fields.

As you can imagine, a properly skilled engineer was quite rare. Pliny was the governor of Bithynia et Pontus, a rich province in northern Asia Minor. Despite its advanced development, Pliny had a hard time finding a qualified engineer. In his letters to the emperor Trajan Pliny asked for permission to build a canal. He was apparently not satisfied with the local engineers and asked Trajan to send an expert. These were in short supply and Trajan put him off, telling him to ask another governor for one.

Vitruvius goes into detail about how to design and build a variety of different structures. The mathematics we would expect in design blueprints are lacking, but they understood loads and stress. Columns had to be of a particular symmetry of diameter to height, both for reasons of structural stability and for beauty.

There were building codes, but these were usually local ordinances and not empire-wide. For example, the city of Rome forbade buildings over a certain height. This is probably why the engineers needed to have some knowledge of the law.

Now to add on a bit more to help your question specifically:

We know about the architects from some works of literature, like Vitruvius whom I have mentioned and Frontinus, who wrote about the aqueducts. We also know about them from tombstones - here is a post I made about surviving blueprints on a tombstone - and their more lasting monuments: the work they left behind.

The architects worked closely with both civil and military officials. Military engineers were very important and were in charge of war machines as well as building projects in the field, such as Caesar's famous bridge over the Rhine.

The organisation of the profession is interesting. It is possible that they were organised much as a skilled civil servant (or a military specialist). I have looked long and hard for evidence of a guild of architects / engineers, and I have yet to find anything. That doesn't mean there wasn't one, as guilds were extremely important in Roman professions. However, it is conceivable that the rarity and mobility of engineers made guilds untenable.