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The notion of "free speech", as we understand it today did not exist in the Roman empire.
The authors you cite probably mean " crimen laesae majestatis", which English Wikipedia translates as "lese majeste". This was a law which was probably introduced under Augustus, and then revoked and re-introduced under various princepces. The first person who revoked it was Vespasian, if I remember correctly. It was not enforced under Marcus, and probably under his predecessors (Trajan, Hadrian, Antonine ...).
So you could say anything about the princeps (or even about his family) during the rule of these liberal princepces. Even in public performances.
Persecution of Christians is a very different matter. There was an established State religion, and established rituals. Refusal to perform them or to participate in them was considered a political act, kind of rejection of the Empire itself. It is not the personal beliefs or speech which was persecuted in this case. But a kind of denial of the supreme authority of the empire. The laws against Christians were not always strictly enforced (or even enforced at all).
A very typical is the attitude of Pliny the younger, who was a governor of a province under Trajan. He did not search for Christians. But those denounced he had to interrogate. If they persisted and refused to perform certain symbolic rituals for established gods, they were executed. (After a third warning and third refusal). Pliny himself says (in a letter to Trajan) that this weird superstition is harmless by itself. But a stubborn denial of the authority and of the established rites deserves a death sentence.
Edit. Interestingly, Dante places Trajan in Paradise, though the proper place for "righteous pagans" according to the official doctrine was the Purgatory. Anyway, Trajan was the only Roman emperor considered a righteous pagan, apparently they do not blame him with his moderate persecution of Christians.
Upvote:2
Free speech is a modern concept. There were no laws in Rome giving a right to speak without being punished. Conversely, there were not too many laws against speech either. There appear to have been some civil remedies against defamation, but the evidence for it is slight and by all appearances it was difficult to sue someone for libel or defamation (see "Roman Law and the Legal World of the Romans" by Andrew M. Riggsby for more information).
We have only fragmentary remains of Roman law to go on and what can be gleaned from literature, so in many cases we don't know exactly what Roman law was or exactly how it was enforced (see Riggsby again). Other than the Augustan dicate against lese majeste, I don't know of any specific laws either allowing or outlawing speech. Even lese majeste itself, was not specifically about speech, but more about any act that would be insulting to the government.
Imperial Rome was a dictatorship and the emperor had the power to arrest or kill people without a trial, so laws were mostly meaningful only in a civil context. When it came to criminal matters, the Roman government apparatus just did whatever it wanted or what the emperor ordered, and this could change drastically from reign to reign. So, for example, Marcus Aurelius had much more moderate policies than some other emperors, but these were not laws; they were imperial policy.
The criminal authorities in Rome, called magistrates, could more or less do what they wanted. So, for example, if someone went around making speeches against the emperor, a magistrate could have him seized and beaten up or killed without any kind of trial. The magistrates sensed the will of the emperor, so when, for example, Marcus Aurelius made speeches about toleration, the magistrates would lighten up and not act against subversives as a result.