There is a certain anachronism in your question to be assuming even the idea of "arrest" existed across time and space. In England there were no police until 1829, and even then it was only established in London (the "bobbies" created by the Metropolitan Police Act of 1829). Before this there were only the Bow Street Runners, who were a private group of less than 10 men (only 6 at the beginning) who had no special legal authority and worked for whoever paid them.
The normal mode of arrest in British law has always been by writ (or "warrant" in modern terminology) which must be produced by indictment before a justice of the peace, meaning a judge. In which case a bailiff or sheriff may then go, writ in hand, and arrest the person.
The law also provided that any person may kill another who attacks him with an intent to rob or kill, and may further arrest and imprison any malefactor who is observed in the commission of a felony, so long as he delivers the person to a justice of the peace or common jail expeditiously. This can be read in the Placiti Coronae (Pleas of the Crown at the King's bench). To quote from the commentary on these pleas by Hale (1778):
If A. a meer private man knows B. to have committed a felony, he may
thereupon arrest him of felony, and he is lawfully in the custody of
A. till he be discharged of him by delivering him to the constable or
common gaol; and therefore if he voluntarily suffers him to escape out
of his custody, tho he were no officer, nor B. indicted, it is felony
It should be noted that historically most bailiffs were private persons anyway and the only thing that distinguished them was possession of a writ of arrest.
Other countries generally always have had similar laws. For example, in pre-Meiji Japan the law was that a person could not arrest another, but you could demand a person's name and insist they report to the nearest police station to answer for a crime. In such cases YOU (the accuser) had to go to the police station immediately as well. To fail to answer such an accusation meant death. Note that you could only arrest a peer in this way. A commoner, for example, could not demand the name of a samurai. If a person was an outlaw in Japan, anybody could kill them freely.
The general pattern in Europe follows the Roman principle venices injuriam, the vengeance of the injured, which allows anyone who is harmed, or even a witness, to raise hue and cry and arrest the perpetrator. For example, in the Republic of Venice anybody could arrest a criminal on the fly. Quoting from Lithgow (1770):
[at Canea]...a young French gentleman, a Protestant, born near
Montpellier in Languedock; who being by chance in company with other
four of his countrymen in Venice, one of them killed a young noble
Venetian, about the quarrel of a courtesan whereupon they flying to
the French ambassador's house, the rest escaped, and he only
apprehended by a fall in his flight, was afterward condemned by the
senators to the galleys during life.
In Islamic Sharia law all people are obliged to enforce the statutes, so perforce, citizen arrests are permitted.