The hope of the Israelite people was that when the Messianic Age or kingdom began to rule, people would not have to ask one another if it was truly here. The Jewish prophets predict a time of prosperity for the Jews and for the Gentile. Not only would Israel be on its land again but a third Temple would stand in Jerusalem (this last part being the only "sign," if you want to call it that, which would mark the arrival of the Messianic Age as unmistakable).
The so-called "sign of Christ's presence" as taught be JWs is significantly flawed. First they claim a Messianic rule that is invisible and entails a series of events contrary to the prosperity foretold and hoped for by the Jewish people since antiquity.
Next it is problematic as the JW eschatology is overrun with mistakes. The Parousia is interpreted as an "insivible" presence instead of a "state" presence or "royal tour of one's kingdom," which is what a "parousia" was in the Hellenistic and Roman world in the first century.
To make matters worse, the Witness make a "sign" out of events which Jesus states are not indicative of his arrival. The discourse regarding the destruction of the Second Temple is used as an eschatological sounding board for events leading to the Second Coming, but each recording of it (Matthew 24, Mark 13, Luke 21) states that wars, earthquakes, famines and plagues are NOT signs that the end is upon them and thus no reason to panic. The JWs mistakenly claim otherwise.
There is a peculiar mention of false preachers associated with the "end of the world" who get it wrong in Luke 21. The text reads:
"Beware that you are not lead astray; for many will come in my name and say: I AM and, 'The time is near!' Do not go after them."-- Luke 21.8, NRSV with footnote reading inserted into text.*
In other words Jesus may here be warning that a group claiming to be Christians will go about using the Divine Name (or a form of it) and proclaim that the time of the end is near. Jesus says about them: "Do not go after them."
*--Generally rendered "I am he!", Luke uses the Greek term EGO EIMI, a euphemistic abbreviation of the Divine Name which literally reads: I AM. The NRSV has "I am" as the alternate reading here due to the fact that the NRSV translation committee felt an equal amount of evidence existed to support a reading that suggested that this false group would be using the Divine Name as part of their attempt to get people to believe their incorrect interpretation. I have placed it here within the text to draw an interesting parallel with the JWs. Regardless of this, even with the original reading intact the text shows that any people proclaiming that the time of the end is near are not to be followed by Christ's disciples.