I don't think "Rights" are ever discussed verbatim in the lore, so we'd have to draw conclusions from what we know of each nation's government.
Ul'dah
A semi-bipartisan government ruled by one Monarch and six citizens making up the Syndicate. Those six members of the Syndicate are the six wealthiest citizens of Ul'dah. The Sultan(a) is from a royal bloodline. (Note: we've had two royal Bloodlines of Ul'dahn monarchy so far)
We can infer there is "freedom" of religion as the Ala Mhigan refugees are not forced to abandon worship of their god Rhalgr. However, citizens must pay a mandatory toll to worship in the houses of Nald'thal.
Crimes against the Sultanate are harshly punished, but the Brass Blades seem to have no restrictions on the extent of their executive powers. They beat, torture, rape, or pillage to achieve their goals. Authority figures either cast a blind eye to this or it is acceptable by law. Meaning law enforcement are above the law.
Speaking of laws, neither the Royalist nor Monetarist parties write laws. The Order of Nald'thal, the religious center of Ul'dah, writes the laws by which the Sultan(a) and Syndicate are bound. However, as one of the leaders of the Order of Nald'thal sits on the Syndicate...
Also, according to the recent 2.4 storyline, apparently the Sultan has the power to overturn these laws if he/she sees fit.
Gridania
Ruled by a the Seedseer Council, a group composed of the eldest Padjal, Gridania is bound by the Laws writ by the Elementals in the Pact of Gelmorra. An agreement made between Man and Elemental which allows Mankind to live among the Wood so long as they care for the Wood's wellbeing. Never take in excess, never bring harm to the Wood or those who live within it, and always give back what you have in abundance. Failure to uphold these basic principles brings on the threat of annihilation from the Elementals and banishment from the Wood for all eternity.
The Duskwights of Gelmorra are (interestingly) not beholden to the Pact of Gelmorra, as they had no desire to live within the Elemental's Wood in the first place.
Limsa Lominsa
Ruled by a Pirate Captain who has won/survived the Trident, a no-holds-barred gauntlet between the pirate crews. The winner of the Trident becomes Admiral of all vessels who sail in Limsa's waters.
Limsa Lominsans are noted as valuing freedom above all else and it appears that (before Merlwyb) there were very very few laws for its citizenry. Pirates will be pirates. However, when Merlwyb Bloefhiswyn took Admiralcy, she united all pirates under a document called the Galadeon Accord. This document, signed by the pirate captains of the Five Seas, united all Pirates under Limsa's banner in the war against the Garlean Empire. Pirates were given the title of Privateers and were paid for their piracy - so long as their piracy was directed towards the Empire and other enemies of the city-state.
Along this same line, there is also an unwritten law called "The Code," by which all Lominsan citizens are bound. This "Code" protects the freedoms of Lominsan citizens so long as they do not attempt to betray or infringe upon the freedoms of their fellow Lominsans.
Know it's not much, but that's about all the lore we have on laws and rights of Eorzeans. Hope this helps! ^^;
Ul'dah
A semi-bipartisan government ruled by one Monarch and six citizens making up the Syndicate. Those six members of the Syndicate are the six wealthiest citizens of Ul'dah. The Sultan(a) is from a royal bloodline. (Note: we've had two royal Bloodlines of Ul'dahn monarchy so far)
We can infer there is "freedom" of religion as the Ala Mhigan refugees are not forced to abandon worship of their god Rhalgr. However, citizens must pay a mandatory toll to worship in the houses of Nald'thal.
Crimes against the Sultanate are harshly punished, but the Brass Blades seem to have no restrictions on the extent of their executive powers. They beat, torture, rape, or pillage to achieve their goals. Authority figures either cast a blind eye to this or it is acceptable by law. Meaning law enforcement are above the law.
Speaking of laws, neither the Royalist nor Monetarist parties write laws. The Order of Nald'thal, the religious center of Ul'dah, writes the laws by which the Sultan(a) and Syndicate are bound. However, as one of the leaders of the Order of Nald'thal sits on the Syndicate...
Also, according to the recent 2.4 storyline, apparently the Sultan has the power to overturn these laws if he/she sees fit.
Gridania
Ruled by a the Seedseer Council, a group composed of the eldest Padjal, Gridania is bound by the Laws writ by the Elementals in the Pact of Gelmorra. An agreement made between Man and Elemental which allows Mankind to live among the Wood so long as they care for the Wood's wellbeing. Never take in excess, never bring harm to the Wood or those who live within it, and always give back what you have in abundance. Failure to uphold these basic principles brings on the threat of annihilation from the Elementals and banishment from the Wood for all eternity.
The Duskwights of Gelmorra are (interestingly) not beholden to the Pact of Gelmorra, as they had no desire to live within the Elemental's Wood in the first place.
Limsa Lominsa
Ruled by a Pirate Captain who has won/survived the Trident, a no-holds-barred gauntlet between the pirate crews. The winner of the Trident becomes Admiral of all vessels who sail in Limsa's waters.
Limsa Lominsans are noted as valuing freedom above all else and it appears that (before Merlwyb) there were very very few laws for its citizenry. Pirates will be pirates. However, when Merlwyb Bloefhiswyn took Admiralcy, she united all pirates under a document called the Galadeon Accord. This document, signed by the pirate captains of the Five Seas, united all Pirates under Limsa's banner in the war against the Garlean Empire. Pirates were given the title of Privateers and were paid for their piracy - so long as their piracy was directed towards the Empire and other enemies of the city-state.
Along this same line, there is also an unwritten law called "The Code," by which all Lominsan citizens are bound. This "Code" protects the freedoms of Lominsan citizens so long as they do not attempt to betray or infringe upon the freedoms of their fellow Lominsans.
Know it's not much, but that's about all the lore we have on laws and rights of Eorzeans. Hope this helps! ^^;