Donald Trump's election was welcomed by many in Russia. But some voices within the country are worried his arrival in the White House will soften America's criticism of Moscow on a host of human rights, corruption and environmental issues.

Vladimir Putin's Russia has been accused by international watchdogs of being anti-democratic, corrupt, of persecuting critics and restricting freedom of speech.

While previous U.S. administrations have been vocal about these allegations, Trump has shown signs he may favor a closer relationship.

Russian parliamentarians broke into a round of applause shortly after Trump's election was announced, and Putin said he is looking forward to a "strong and enduring relationship" with the president-elect.

For his part, Trump spoke favorably about Putin during the campaign, and picked ExxonMobil CEO Rex Tillerson — who has close links to Putin — is his nomination for secretary of state.

U.S. intelligence officials have alleged that Putin was personally involved a covert campaign to influence the vote — with the ultimate goal of electing Trump.

Voices that champion civil society causes, like non-profits and NGOs, fret that all this suggests Trump's America will be less focused on pointing out potential abuses or issues that arise within Russia.

"The United States has been one of the critical voices of what’s going on in Russia," said Natalia Taubina, head of the Public Verdict Foundation, a Moscow non-profit organization offering legal assistance to rights-abuse sufferers. "My concern is that [under Trump] the U.S. will pay less attention to the human rights situation in Russia and issues like torture."

"Russia does not like to be criticized, let alone humiliated, in the public sphere"

Organizations like Taubina's have come under severe pressure from the Russian government in recent years.

Laws were introduced in 2012 that label groups as "foreign agents" if they receive money from international donors and engage in broadly defined political activity.

According to Human Rights Watch, the term foreign agent is akin to being classed as a spy or traitor.

Russia's Ministry of Justice has listed more than 145 groups as foreign agents since 2012 — some of whom have faced court cases, fines and costly audits — while 28 have closed down rather than bear the label, HRW said earlier this month.

For Anton Pominov, director general at Transparency International Russia, the law is designed to "make a little bit harder the life of the organizations that are not dependent on the Russian government."

Grigory Melkonyants of election monitor Golos listens to a judge's verdict regarding a fine imposed under the foreign agents law in 2013. AP

Dmitry Lisitsyn, director of environmental non-profit Sakhalin Environment Watch, agreed.

Lisitsyn told NBC News he believes the main purpose of the foreign agents law is to "frustrate the people, to demotivate them, to undermine their dedication to the work that they are doing."

He added that the main beneficiaries of the law are the Kremlin and the FSB domestic intelligence service, the latter of which Lisityn said are "deep in their paranoia" on the issue of foreign funding for non-governmental organizations.

The Obama administration has been critical of these developments. In 2013, Obama's national security adviser, Susan Rice, hit out at Russia for its "systematic efforts to curtail the actions of Russian civil society."

But how would a Trump presidency and a potentially closer relationship with Moscow change this?

Russian President Vladimir Putin NATALIA KOLESNIKOVA / AFP - Getty Images

In his recent annual address, Putin argued that charities and nonprofits should be given "substantive support," Russia's state-run TASS news agency reported. Yet few of the organizations NBC News spoke with had confidence in these pledges.

Amendments made to the foreign-agents law this year have only led to it becoming "much more strong and harmful," said Lisitsyn, whose organization was forced to return a donation by the Leonardo DiCaprio Foundation last year as it would have counted as foreign funding.

On top of the foreign agents law, a number of U.S. and international foundations deemed to be undermining Russia have been banned from operating within the country under separate 2015 legislation targeting "undesirable" organizations.

USAID was also barred from operating in Russia back in 2012.

However, figures for 2015 published on the USAID website reveal the State Department made multiple donations totaling $4.95 million to unspecified NGOs that deal with Russia via National Endowment for Democracy grants.

Sergei Nikitin, head of the Russian branch of Amnesty International, called for the country so abolish its foreign agents law in November. AFP/Getty Images

Taubina said her organization doesn't accept American money, even though it does receive donations from other foreign sources as well as from the Russian government.

What's more important for her group is the potential loss of a strong supportive voice for its cause. Although many consider Russia deaf to such interventions, Taubina disagrees.

"It's important to continue this criticism as it helps us in our work," she said.

This is a point of view shared by Leonard Benardo, regional director for Eurasia at the Open Society Foundations, one of the international organizations forced shut down its Russia operations in 2015.

"Russia does not like to be criticized, let alone humiliated, in the public sphere," Benardo said. "They don’t like bad press. They don't appreciate to find themselves in situations where they are harangued or criticized."

Benardo added that external criticism does have its limits and it is hard to predict precisely how U.S.-Russia relations will develop. But he fears that "notions of human rights and democratic practice would be very much put to one side" with other strategic U.S. foreign policy issues dominating over values.

Vladimir Putin presents ExxonMobil CEO Rex Tillerson with a medal in 2012. Mikhail Klimentyev / Kremlin Pool Photo via AP

Benardo said while that would be a "boon" to Putin and the Russian government, "it would be a manifest disservice to Russian civil society."

Anna Sevortian, executive director of the E.U.-Russia Civil Society Forum, told NBC News she was hopeful that the incoming U.S. administration "stays very principled" on issues of human rights and democracy and "it's not just economics, energy, security talk."

For Taubina, however, the worry is that such a situation as described by Benardo comes to pass.

"What we are witnessing here, what we are experiencing here, can be set in very simple formula. As long as there is international attention, we are more or less secured," she said. "If there is no international attention, international assessment … towards what’s going on in Russia, we are much less safe with our mission."