"Citizens must be empowered to help formulate the laws under which they live," she said, adding: "They need to know that their investments of time, money and intellectual property will be safeguarded by the institutions of government."
Earlier, she was pressed by journalists on the failure of the Russian authorities to find the killer of journalist Anna Politkovskaya and on the imprisonment of ex-tycoon Mikhail Khodorkovsky.
"I think all of these issues - imprisonments, detentions, beatings, killings - is something that is hurtful to see from the outside," she told Echo of Moscow radio.
Hillary Clinton talks about democracy and journalism on a Moscow radio station
"Every country has its criminal elements, people who try to abuse power. But in the last 18 months... there have been many of these incidents.
"I think we want the government to stand up and say this is wrong."
Prior to the round-table discussion with students, Ms Clinton attended the unveiling of a statue of US poet Walt Whitman on the university's campus.
Whitman is often labelled the father of free verse, as well as being a prominent figure adopted by the gay pride movement.
"Just as Pushkin and Whitman reset poetry, we are resetting our relations for the 21st Century," she said.
A statue of the Russian Romantic poet Alexander Pushkin was placed at George Washington University, in Washington, in 2000.
Ms Clinton will be the first US secretary of state to visit the city of Kazan, capital of Tatarstan, a Muslim-majority republic which is often hailed as a model of multicultural tolerance.
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