American families will be monitored by U.S. social services for three years after adopting a child. After that, the agency that handled the adoption will monitor the home until the child reaches 18, reporting to Russian authorities all instances of abuse, as well as the termination of an adoption or re-adoption by another family.This article says the monitoring will include 4 home visits.
Another change will see the children keeping dual citizenship until they reach 18, whereas previously they were stripped of their Russian citizenship immediately after adoption.
There seems to be a bit of a disagreement, though, about what the monitoring of American families will entail, and how much authority the Russian government will have. According to the background briefing from the U.S. State Department, the monitoring will be exclusively through U.S. adoption agencies, with the Russian Government not having any right to go into a family’s home in the United States. But this Voice of Russia article quotes Foreign Minister Sergei Lavrov as saying that the agreement ensures "there is an access of the Russian government to a kid if need be." An in this article, we have both sides dueling accounts:
Lavrov praised the deal for including "guarantees and safeguards for both sides" and said Russian diplomats would be given access to adoptive parents.
But a US official, speaking on condition of anonymity, denied that any US or Russian government officials would be allowed into homes for checks. The official said that monitoring would be conducted by adoption providers
Sounds to me, though, that the upshot of the monitoring is what is already pretty standard in international adoption for post-placement monitoring -- that it's going to be the obligation of the agency, and if they fail to satisfy Russia, then Russian can yank their authorization to operate in Russia. And that means if the agency fails to make the child available to Russian officials or fails to get the adoptive family to cooperate, they'll lose that authorization. So Russia may not have a "right" of access, but they can put all kinds of pressure to bear on an agency to get that access for them.
More information -- the State Department FAQ & the USCIS FAQ.