The demonstrations were called to mark the 38th anniversary of the banning of political parties.
The authorities have banned the protests and arrested five organisers.
A BBC correspondent in the main city, Manzini, says there are more security agents than civilians on the streets.
The BBC's Nomsa Maseko saw a group of riot police marching up the street singing: "You will get arrested if you dare."
People can only approach the venue of the planned protests if they are on their own, she says.
Mario Masuku, head of the banned opposition People's United Democratic Movement (Pudemo), has been put under house arrest, his deputy told the BBC.
Most of the protesters are said to be in the capital, Mbabane, and are unable to get to Manzini, a 45-minute drive away.
The demonstrations are supposed to last for three days.
Last month, thousands of civil servants held the biggest march in Swaziland for several years, to protest at a pay freeze and demand that the government resigned.
The protesters are keen to stress that they do not want to oust King Mswati III but they want a constitutional monarchy.
The king - who has 14 wives - has been accused of living a lavish lifestyle, while hundreds of thousands live in poverty.
Swaziland has a population of only 1.4 million but 40% of them are unemployed, and 70% of the population is living on less than $1 (75p) a day.
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