The Palestinian prisoners seek better conditions, including more contact with relatives, and an end to Israel's practice of detentions without trials.
Meanwhile on Tuesday, Israeli Public Security Minister Gilad Erdan vowed not to negotiate with Palestinian hunger strikers, saying that Barghouti had been moved to another prison and placed in solitary confinement.
The 1,200, of about 6,500 Palestinians held by Israel for terrorism, joined the open-ended protest, the largest such strike in five years.
Israeli forces clashed with Palestinian protesters standing in solidarity with their compatriots embarking on hunger strikes in Israeli prisons.
There are fears that the hunger strike - observed by 1187 detainees, according to Israel's prison service - could fuel tensions across the Palestinian areas.
The strikes fall in tandem with Palestinian Prisoners Day and also marks 50 years since the Six Day War between Israel and Egypt in 1967 - where Israel were victorious and captured the Gaza Strip.
One cage held two Palestinians in Israeli prison uniform.More news: Wasn't it Trump who begrudged Obama his golf trips?
The essay asserted that "Israel has established a dual legal regime, a form of judicial apartheid, that provides virtual impunity for Israelis who commit crimes against Palestinians, while criminalizing Palestinian presence and resistance".
The NGO Palestinian Prisoners' Club said 1,500 people were taking part.
"These are terrorists and incarcerated murderers who are getting exactly what the global law requires", Erdan said in an interview with Israel's Army Radio.
Israeli prison officials reportedly accused Barghouthi of using his wife to "smuggle" the article out of prison and to the New York Times. However, Palestinians from the West Bank need an entry permit to enter Israel and the Israeli authorities often deny such permits. The hunger strike was orchestrated by Marwan Barghouti, who is also the head leader of this movement.
"I have been both a witness to and a victim of Israel's illegal system of mass arbitrary arrests and ill-treatment of Palestinian prisoners".
The strike was called by imprisoned Fatah party leader Marwan Barghouti, who is serving five life terms for his involvement in the killing of Israelis earlier this century. "It is to be emphasized that the (Prisons Service) does not negotiate with prisoners". Building new settlements in the occupied territories is considered a violation of global law.
Among the prisoners' chief demands is the installation of telephones in each prison, more frequent family visits, better healthcare and improved conditions for female prisoners. Erdan continued his remarks on Israeli public radio by describing the prisoners as "terrorists".