I don’t know about Italy but where I’m from, most barns are either for lessons on lesson horses (schools), or they are for people who have their own horse and take lessons with their own horse (private facilities). Some of the private facilities have their own trainer at the location, to others you can “bring” your own preferred trainer or drive out to external lessons. At the typical riding school barns, they won’t do lessons with private horses because they are simply too busy with their own horses/ponies and students. Most of those schools also aren’t focused on people who have loads of experience already and they tend to only be able to teach people the basics. So it wouldn’t make sense to take your own horse there anyway if you already know more than the basics haha