The American Civil Liberties Union says state legislators are proposing various restrictions on local authorities' use of the technology.
Concerns mounted after the Federal Aviation Administration began establishing safety standards for civilian drones, which are becoming increasingly affordable and small in size.
Some police agencies have said the drones could be used for surveillance of suspects, search and rescue operations, and gathering details on damage caused by natural disasters.
In Montana, a libertarian-minded state that doesn't even let police use remote cameras to issue traffic tickets, Democrats and Republicans are banding together to back multiple proposals restricting drone use. They say drones, most often associated with overseas wars, aren't welcome in Big Sky Country.
"I do not think our citizens would want cameras to fly overhead and collect data on our lives," Republican state Sen. Matthew Rosendale told a legislative panel on Tuesday.
Rosendale is sponsoring a measure that would only let law enforcement use drones with a search warrant, and would make it illegal for private citizens to spy on neighbors with drones.
The full Montana Senate endorsed a somewhat broader measure Tuesday that bans information collected by drones from being used in court. It also would bar local and state government ownership of drones equipped with weapons, such as stunning devices.
The ACLU said the states won't be able to stop federal agencies or border agents from using drones. But the Montana ban would not allow local police to use criminal information collected by federal drones that may be handed over in cooperative investigations.
The drones could be wrongly used to hover over someone's property and gather information, opponents said.
"The use of drones across the country has become a great threat to our personal privacy," said ACLU of Montana policy director Niki Zupanic. "The door is wide open for intrusions into our personal private space."
Other state legislatures looking at the issue include California, Oregon, Texas, Nebraska, Missouri, North Dakota, Florida, Virginia, Maine and Oklahoma.
A Missouri House committee looked at a bill Tuesday that would outlaw the use of unmanned aircraft to conduct surveillance on individuals or property, providing an exclusion for police working with a search warrant. It drew support from agricultural groups and civil liberties advocates.
"It's important for us to prevent Missouri from sliding into a police-type state," said Republican Rep. Casey Guernsey of Bethany.
A North Dakota lawmaker introduced a similar bill in January following the 2011 arrest of a Lakota farmer during a 16-hour standoff with police. A drone was used to help a SWAT team apprehend Rodney Brossart.
Its use was upheld by state courts, but the sponsor of the North Dakota bill, Rep. Rick Becker of Bismarck, said safeguards should be put into place to make sure the practice isn't abused.
Last year, Seattle police received approval from the Federal Aviation Administration to train people to operate drones for use in investigations, search-and-rescue operations and natural disasters. Residents and the ACLU called on city officials to tightly regulate the information that can be collected by drones, which are not in use yet.
In Alameda County, Calif., the sheriff's office faced backlash late last year after announcing plans to use drones to help find fugitives and assist with search and rescue operations.
David A. Lieb contributed to this report from Jefferson City, Mo.