Well, it depends on the genre. There was the direct patronage model, where a composer worked at the discretion of a rich patron (like Haydn, for instance). But there were also more indirect methods as well. A musician could work for a church, for instance, and not necessarily answer to any noble person directly. One could also work on the operatic circuit, in which case one worked under a controlling musical director (called an "impressario"), rather than answer directly to any of the noble patrons of the theatre.

I could possibly help to shed light on this latter situation, as I'm most familiar with the process of opera rehearsing. Though it will have to wait for a few days until after I finish moving and I unpack my books. But I do want to make sure before I do that that this is actually a part of the story you are interested in. Because if you are interested more in how the direct patronage model worked, I know less about that. Just letting you know at this stage that there were many diverse circumstances in which musicians worked during the eighteenth century!

I can also at this stage point to one of the most fascinating books on the subject, in case you are at all interested in just reading some scholarship on the subject. William Holmes's Opera Observed: Views of a Florentine Impressario has more information on the process of producing an opera than you would ever want to know! Really fascinating stuff.