How free was independent thought in Roman society?

That's going to be a though one. Because Roman society differed very much from ours.

Roman society was highly stratified. Senate on top, poor people of less than 5th class firmly at the bottom. The Romans didn't think this was the way it should be. They KNEW it.

Everybody was easily recognizable the way they clothed themselves. You were punishable by law if you wore for example the maroon shoes of a senator without being one. It's difficult to imagine a guy being prosecuted for wearing the same shoes as the president, right?

A senator would be utterly surprised when he hears about tax-evasion. Why? In those days you wanted to pay tax. The more the better. That was the the only purpose of being rich: to show off how much you could afford to spent on the public good.

There wasn't equal representation. The richer you were, the more of the pie would be yours to enjoy. In order to get to the pie, you had to work very hard in achieving glory and doing your duty. Even if you were a lazy person or yellow livered what ever (as Cicero actually was) you would hide that fact as much as possible.

Exactly the same applies for freedom of thought. Just like today, it was completely free - provided it fitted into the framework of what was common knowledge. A rich man would always pursue personal glory and do his duty. Dying while doing his duty, if need be.

The only real limitation you can argue is that one had to consider the emperor in imperial times. Lots of emperors could be, if not amused, rather difficult.

Today is not that much different. Read the book: "Vote for Caesar" by Peter Jones. Gives you exactly the inside idea of what you want to research