Minority Shi'ites have staged sporadic protests in the province for years. The latest wave of demonstrations coincided with the Arab Spring uprisings that began in North Africa in 2011.
The www.rasid.net news website identified him as Ali al-Rubh, 26, and said his father, Sheikh Rida al-Rubh, has been leading contacts with the Saudi Interior Ministry to restore calm to the town of Awamiya, where most of the protests have taken place.
It said the convict plans to appeal the ruling.
"This was the first death sentence of its kind since protests marches began in Qatif three years ago," sabq.org, another Saudi news website, said.
The Arabic daily Okaz said on its website that the man, whom it did not name, was found guilty of opening fire at security forces in the towns of Tarout and Darin, both east of the Qatif governorate, which has been at the heart of recent protests. The newspaper made no reference to any casualties from the attack.
The Jeddah court also found him guilty of buying weapons and harboring a wanted man and rioters, it said.
At least 21 people have been shot dead in the Eastern Province since early 2011. Most Saudi Shi'ites live in the region and some say they suffer discrimination in the Sunni-ruled kingdom, a charge authorities deny.
The rasid.net website said that around 300 people are still held on charges linked to the protests. REUTERS