Egyptian opposition activists have vowed to stage a sit-in in Cairo in a protest against the president's move to grant himself extensive new powers.
Crowds of people gathered in Cairo's Tahrir Square on Friday but had largely dispersed by morning, with reports of police firing tear gas nearby.
More than 100 people have been injured in clashes across the country.
President Mohammed Mursi says he is leading Egypt to "freedom and democracy".
The decree, issued on Thursday, bans challenges of his decisions and says no court can dissolve the constituent assembly, which is drawing up a new constitution.
It also opens the way for a retrial of people convicted of killings during Egypt's 2011 uprising which toppled Mubarak.
The US has voiced concern about developments.
'Mursi is Mubarak'
Both supporters and critics of the president demonstrated across the country on Friday.
Critics are calling for the constitutional declaration to be reversed, and for the dissolution of the constituent assembly.
Overnight, some 20 white tents were pitched in the middle of Cairo's totemic Tahrir Square, the scene of the unrest which ousted long-time President Hosni Mubarak last year.
Protest organisers said more than 20 different groups had joined a week-long sit-in against Mr Mursi's reforms, saying the new president is becoming as much of a dictator as Mubarak.
"We are involved in a sit-in because the goals of the revolution have yet to be achieved," said Mira Daniel, whose brother, Coptic activist Mina Daniel, was killed in the 2011 uprising.
"We are here because they took a decision to make us die in the square, so we are here to die in the Square. What's happening now is death."
However, most people left the square overnight, with dozens of messages on Twitter saying protesters were forced to abandon the camp after being targeted with tear gas.
By Saturday morning, small numbers of people were beginning to gather in the square again. A larger protest is planned for Tuesday.
Separately on Friday evening another set of anti-Mursi protesters clashed with police near the interior ministry off Tahrir Square.
Police fired tear gas and demonstrators threw petrol bombs.
Chants of "Mursi is Mubarak... revolution everywhere" rang out.
More than 100 people were injured in violence as offices of the president's Muslim Brotherhood party were reportedly attacked in the cities of Port Said, Suez and Ismailia, while clashes between rival demonstrations took place in Alexandria.
Defending his decision at a rally at Cairo's presidential palace on Friday, Mr Mursi said he was the guardian of political, economic and social stability, but that he wanted to see a "genuine opposition, a strong opposition".
"I don't like, want or need to resort to exceptional measures, but I will if I see that my people, nation and the revolution of Egypt are in danger," he told his supporters.
US state department spokeswoman Victoria Nuland said on Friday that the decree had "raised concerns" in the international community.
She said Egypt's revolution had aimed "to ensure that power would not be overly concentrated in the hands of any one person or institution".
In recent days, Mr Mursi had been praised by the US and other nations for his efforts in brokering the ceasefire in Gaza.
The declaration also gives the 100-member constituent assembly two additional months to draft a new constitution, to replace the one suspended after Mubarak was overthrown.
The rewrite of the constitution, which was meant to be finished by December, has been plagued by legal complaints questioning the make-up of the constituent assembly.
Once completed, the document is due to be put to a referendum. If it is approved, legislative elections will be held two months later.
The BBC's Jon Leyne in Cairo says one element of the decree that is likely to be popular is the order to reopen investigations into Mubarak, his sons and other former senior officials.
The ousted leader is serving a long jail term for ordering the killing of protesters in 2011, but others in his administration were acquitted, a cause of deep resentment in Egypt, says our correspondent.