Government policy states, in part, that "[a] work that is a United States Government work, prepared by an officer or employee of the United States Government as part of that person's official duties, is not subject to copyright in the United States and there are no U.S. copyright restrictions on reproduction, derivative works, distribution, performance, or display of the work." This is consistent with the Copyright Act as well. Images on the White House Flickr photostream, however, now contain this restrictive legend: "This official White House photograph is being made available only for publication by news organizations and/or for personal use printing by the subject(s) of the photograph." (Example here.) It then goes on to prohibit the commercial use of the image in any way that suggests the endorsement of the subject.
Oddly enough, the Flickr page also contains a link to the Government Policy that disclaims a copyright interest in images that are United States Government works -- as most of the White House photostream images are.
Even more oddly, the restrictive legend is not the same for all images. This image of the President meeting with former president George H.W. Bush removes the word "only" from the restrictive legend, which changes its meaning considerably, from a restrictive one to an explanatory one: "This official White House photograph is being made available for publication by news organizations and/or for personal use printing by the subject(s) of the photograph." The lack of an "only" there suggests that others can use the image as well. Perhaps it's a partisan thing.
There is of course a tension between the rights that the creator of an image has in that image (which are addressed by copyright law) and those that the subject of the image has in that same image (which are addressed by publicity laws in most states that give individuals some rights over how their images are used for advertising or other purposes). This was highlighted in a case brought two years ago by the family of a Texas teenager whose picture was taken by a photographer and posted on Flickr with a Creative Commons Attribution License. The family sued when Virgin Mobile used the image in ads for its mobile phone service in Australia. The issue there was really not one of copyright, but of publicity.
The President recently experienced the same sort of situation, when the Weatherproof clothing brand used an image of the President wearing a Weatherproof brand jacket on a billboard ad in Times Square.
Perhaps what the White House means to say is best expressed in the second half of the restrictive legend it now posts with its Flickr images, which states that the images "may not be used in commercial or political materials, advertisements, emails, products, promotions that in any way suggests approval or endorsement of the President, the First Family, or the White House." That legend is grounded in publicity law, not copyright, and is not inconsistent with the government disclaimer of copyright. It would perhaps be the better limitation for the White House to emphasize.