All perfect praise be to Allah
You should read rajm-wikipedia page, their is some information in fiqh section. Also it must be kept in mind that rajm/stoning was not a new kind of punishment, it was an old punishment which was also practiced by Jews of that time(it was present in their scriptures too), so there was nothing new to explain. Though we find how some RAJAM punishments were executed in ahadith. Also have a look at stonning-wikipedia to have a look at stonning history.
Below is an incident of stoning a jew in era of prophet Muhammad peace be upon him
Narrated Ibn 'Umar: A Jew and a Jewess were brought to Allah's Apostle
on a charge of committing an illegal sexual intercourse. The Prophet
asked them. "What is the legal punishment (for this sin) in your Book
(Torah)?" They replied, "Our priests have innovated the punishment of
blackening the faces with charcoal and Tajbiya." 'Abdullah bin Salam
said, "O Allah's Apostle, tell them to bring the Torah." The Torah was
brought, and then one of the Jews put his hand over the Divine Verse
of the Rajam (stoning to death) and started reading what preceded and
what followed it. On that, Ibn Salam said to the Jew, "Lift up your
hand." Behold! The Divine Verse of the Rajam was under his hand. So
Allah's Apostle ordered that the two (sinners) be stoned to death, and
so they were stoned. Ibn 'Umar added: So both of them were stoned at
the Balat and I saw the Jew sheltering the Jewess.
8:82:809, See also: Sahih Bukhari 6:60:79, and Sahih Bukhari 4:56:829
and this is also similar
Narrated 'Abdullah bin 'Umar : The Jew brought to the Prophet a man
and a woman from amongst them who have committed (adultery) illegal
sexual intercourse. He ordered both of them to be stoned (to death),
near the place of offering the funeral prayers beside the mosque."
Sahih Bukhari 2:23:413
Secondly, lets have a look at some other ahadith about stoning execution:
It was narrated that Abu Hurayrah and Zayd ibn Khaalid al-Juhani (may
Allaah be pleased with them both) said: The Messenger of Allaah (peace
and blessings of Allaah be upon him) said: “… Go tomorrow, O Unays, to
this woman and if she admits (to adultery) then stone her.” He said: I
went to her the next day and she admitted it, so the Messenger of
Allaah (peace and blessings of Allaah be upon him) ordered that she be
Narrated by al-Bukhaari, 2575; Muslim, 1698.
It was narrated from Jaabir that a man from (the tribe of) Aslam came
to the Prophet (peace and blessings of Allaah be upon him) when he was
in the mosque and he said, “I have committed adultery.” The Prophet
(peace and blessings of Allaah be upon him) turned his face away from
him. The man came to that side to which the Prophet had turned his
face, and testified against himself four times. The Prophet (peace and
blessings of Allaah be upon him) said, "Are you mad?” He said, “No.”
The Prophet said, “Are you married?" He said, "Yes.” So he commanded
that he be stoned in the Musalla (Eid prayer place). When the stones
troubled him, he ran away, until he was caught in al-Harrah and
Narrated by al-Bukhaari, 4969; Muslim, 1691.
Lets just stay with these here, there are also some other which have similar meanings.
It is clear form these incidents that rajam is to be done in public place, as it mentions prayer ground besides mosque in hadith.
And from the last hadith it is clear that he ran, meaning he was not burried half in earth(as some fiqh say so, i don't know about their evidence for this), only than he could had ran.
In hanafi school of thought In cases of self-confession, the accused is neither bound nor partially buried and allowed to escape during stoning. In the other case, according to Hanafi legal texts, the accused is bound and partially buried inside a bit in standing position so he or she cannot escape, and then stoning must be performed till he or she dies. From stonning section here
Lastly there is no rocket science in meaning of stonning to death, it means, the person will be stoned until he dies(same as cutting hand of thief means just cut his hand of... no need to precisely mark where to cut and so on. Cut the hand and that's it.)! Also it does not need to be mentioned that the punishment is to be carried out by government.
IMPORTANT NOTE: I am not a mufti and all of this is based on my research and my understanding which can be wrong and must not be trusted. Pleas correct me if you find anything wrong in it.
Allah knows best