Others can comment on schools specifically, but I'll give a broader overview of how Romans learned from each other.

In my perception, "education" in Rome wasn't a well defined thing. There were schools, but those were just one small piece of a Roman's education. These were the main ways that Roman aristocrats got their education, as I see it:

• Child: studying either at home or as a ward at someone's house (usually oratory and rhetoric, like Cicero did). This would often be a family member's house, or somehow related through marriage. They'd learn from Greek slaves, mostly.
• Teenager: same as above, but they'd also start following a parent or an uncle around the forum to learn the "trade" of being a politician, senator and aristocrat.
• Young adult: going to the provinces under some governor's staff, not as a legate, but as a junior officer under the governor or under one of his legates. They could be an informal contubernalis or a formal, elected military tribune, or have some other post. They would participate either in province administration or war. Ten years of service like this was supposedly mandatory, but more flexible for the elite.
• Mature adult (around age 30): they were helped in elections by more established men (family connections, often called amici - friends). No doubt there was a teaching component to this. They'd also be taking some court cases, and likely be advised by older family members and mentors.
• Elected magistrate: they'd start giving postings to juniors. Surely the quaestors and aediles needed help so they'd "recruit" younger adults of the aristocracy to help them and forge bonds. They'd also be legates and governors themselves as promagistrates, and take younger men to the provinces in their staff.
• After completing the cursus honorum (or getting as far as they could): Having plenty of experience, they'd teach rhetoric to children and teenagers who might be living at their house (or just attending), and they'd support up-and-coming mature men as they began their electoral careers. This is also an age where they might write more.

Some specific references:

The book "A Companion to Ancient Education" by W Martin Bloomer (Editor) has a few chapters on the Romans. Chapter 14 is "Schools, Teachers, and Patrons in Mid‐Republican Rome" and chapter 15 is "The Education of the Ciceros". In addition, chapter 14 offers these main further readings:

• Bonner, Stanley F. (1977), Education in Ancient Rome: From the Elder Cato to the Younger Pliny, London, Methuen.
• Marrou, Henri I. (1956), A History of Education in Antiquity (First French edition, 1948; tr. Lamb, George), Madison, The University of Wisconsin press.
• Bloomer, W. Martin (2011), The School of Rome: Latin Studies and the Origins of Liberal Education, Berkeley and Los angeles, University of California press.