What stops a modern Jew from practising stoning?

Question

What stops a modern Jew from practising stoning? why don’t observant Jews keep the laws of death penalties as prescribed in the Torah?

Answer 1

The death penalty is not carried out nowadays due to the fact that we no longer have a standing Temple nor do we have an actual Sanhedrin with semikhah.

(Semikhah is an unbroken chain of Rabbinic ordination going back to Moses our Teacher.)

But even when we did have these things, the Mishnah tells us that the death penalty was rarely carried out - so rarely in fact that if a court issued the death penalty once in seven years they were called "bloodthirsty." (Makkot 1:10) According to other opinions (mentioned there as well), it was even once in seventy years.

Eventually, the death penalty will be brought back once the Temple is rebuilt and when the Sanhedrin is reinstated, when G-d willing the Messiah comes soon, may it be in our days.

Further reading: Talmud, Sanhedrin 2a

Answer 2

The canon of Jewish law is more expansive than the text of the written Torah. The judicial process is fleshed out in the oral law and in it, we learn that there are many steps and requirements, and through later writings we understand that the conditions for the application of capital punishment no longer exist.

Here is a nice discussion of capital punishment and it mentions some fo the details and limitations.