In short, the argument you heard is sound. In John 18:31, the Jewish leaders make it clear they are unable to execute anyone by Roman law:
So Pilate said to them, "Take Him yourselves, and judge Him according to your law." The Jews said to him, "We are not permitted to put anyone to death," —John 18:31 NASB
The Popular and Critical Bible Encyclopaedia and Scriptural Dictionary, Fully Defining Explaining All Religious Terms, Including Biographical, Geographical, Historical, and Archaeological Doctrinal Themes, Volume 2 by Herbert Lockwood Willett confirms this on page 951:
But the Sanhedrin could only pass, not execute, the death sentence.
Here is a similar sentiment from Barnes' Notes on the Bible:
The Jews themselves say that the power of inflicting capital punishment was taken away about 40 years before the destruction of the temple
And the same from Bengel's Gnomen:
Jewish history accordingly tells us that on that very year, the fortieth before the overthrow of the city, the power was taken from them.
A list of arguments is provided by the Cambridge Bible for Schools and Colleges:
On the one hand we have (1) this verse; (2) the statement of the Talmud that 40 years before the destruction of Jerusalem the Jews lost this power; (3) the evidence of Josephus (Ant. xx. ix. 1; comp. xviii. i. 1; xvi. ii. 4, and vi.) that the high priest could not summon a judicial court of the Sanhedrin without the Procurator’s leave; (4) the analogy of Roman law.
Matthew Poole's Commentary provides another argument:
We are assured by such as are exercised in the Jewish writings, that the power of putting any to death was taken away from the Jews forty years before the destruction of Jerusalem. Some say it was not taken away by the Romans, but by their own court. They thought it so horrid a thing to put an Israelite to death, that wickedness of all sorts grew to such a height amongst them, through the impunity, or too light punishment, of criminals, that their courts durst not execute their just authority.
The Pharisees may have risked the execution of the woman in question because it was not anything the Roman government would take notice of. They could not stone Jesus outright because he was a figure the Romans were aware of. As for the woman accused of committing adultery, this website has a fairly thorough, straightforward explanation:
Under Roman rule, the power to impose capital punishment, including by stoning, had been taken away from all Jewish authorities. Only a Roman tribunal could impose the death penalty. That is why even after Jesus was eventually arrested and condemned to death by the Jewish Sanhedrin, they didn’t stone him immediately themselves; he had to be taken before the Roman governor, Pilate, to actually impose the death penalty. And when it was imposed he was killed using the Roman method — crucifixion– not stoning. The Jewish leaders had no legal authority to put him to death.
The rest of the article appears to be the writer's personal opinions on the state of the Church today, but the logic in the above paragraph is sound.
To specifically address the commonality of illegal lynchings at the time, it is reasonable to assume they were a rare occurrence. On the contrary, it was seen as an abhorrent thing. Poole's Commentary above summarizes the Jewish people's attitudes towards the execution of Israelites. In addition, it is common knowledge that the Romans actively oppressed the Israelites. They would not risk the wrath of Rome to perform an action that they already hated.
This is completely unrelated to the question, but I found it while researching.
It's pretty interesting. Jesus and His World: An Archaeological and Cultural Dictionary by John J. Rousseau provides a description of how extralegal stonings such as this one occurred in Jesus' time. Here is an excerpt from page 264:
In the time of Jesus, the [Jewish] legal procedure was probably closer to that of the Mishnah. It allowed for new evidence in favor of the offender until the last minute and for confession before death. According to this text the condemned was first stripped of his garments and thrown down from a platform nine feet high by the first witness. If he survived the fall, the second witness cast a heavy stone on his chest and, if death did not occur at this point, all those present stoned him. After stoning, the body was hanged on a tree.