Found some interesting reading on the subject:
Viacom Inc. v. YouTube, Google Inc.
On March 13, 2007, Viacom filed a lawsuit against YouTube and its corporate parent Google for copyright infringement seeking more than $1 billion in damages. The complaint was filed in the U.S. District Court for the Southern District of New York. Viacom claims the popular video-sharing site was engaging in "massive intentional copyright infringement" for making available a contended 160,000 unauthorized clips of Viacom's entertainment programming. Google lawyers say they are relying on the 1998 Digital Millennium Copyright Act to shield them from liability.
On June 23, 2010, U.S. District Judge Louis Stanton granted summary judgment in favor of YouTube. The court held that YouTube is protected by the safe harbor of the DMCA. Viacom has said that it will appeal before the U.S. Court of Appeals for the Second Circuit as soon as possible.
Also:
Title II: Online Copyright Infringement Liability Limitation Act
DMCA Title II, the Online Copyright Infringement Liability Limitation Act ("OCILLA"), creates a safe harbor for online service providers (OSPs, including ISPs) against copyright liability if they adhere to and qualify for certain prescribed safe harbor guidelines and promptly block access to allegedly infringing material (or remove such material from their systems) if they receive a notification claiming infringement from a copyright holder or the copyright holder's agent. OCILLA also includes a counternotification provision that offers OSPs a safe harbor from liability to their users upon notice from such users claiming that the material in question is not, in fact, infringing. OCILLA also provides for subpoenas to OSPs to provide their users' identity.
In further research it can be determined that a social networking site may be considered an OSP (Online service provider):
The Online Copyright Infringement Liability Limitation Act (OCILLA) portion of the U.S. Digital Millennium Copyright Act has expanded the legal definition of online service in two different ways for different portions of the law. It states in section 512(k)(1):
(A) As used in subsection (a), the term "service provider" means an entity offering the transmission, routing, or providing of connections for digital online communications, between or among points specified by a user, of material of the user’s choosing, without modification to the content of the material as sent or received.
(B) As used in this section, other than subsection (a), the term "service provider" means a provider of online services or network access, or the operator of facilities therefore, and includes an entity described in subparagraph (A).[1]
These broad definitions make it possible for a large number of web businesses to benefit from the OCILLA.
It seems that if there are any claims of infringement the owner of the site must comply and take the said material offline. In cases where there was prosecution the sites either refused to the take down or had repeated requests ignored.