As far as I know, any museums that are associated with the federal government have to adhere to the following rule when it comes to downsizing the parts of their collection that are not being exhibited. I have never been to a Smithsonian sale, but I have a close friend who has twice. I found the statute that explains this:
"(h) CONVEYANCE OF MUSEUM OBJECTS.—The Secretary may convey museum objects that the Secretary determines are no longer needed for museum purposes, without monetary consideration but subject to such terms and conditions as the Secretary considers necessary, to private institutions exempt from Federal taxation under section 501©(3) of the Internal Revenue Code of 1986 (26 U.S.C. 501©(3)) and to non-Federal governmental entities if the Secretary determines that the recipient is dedicated to the preservation and interpretation of natural or cultural heritage and is qualified to manage the property, prior to any conveyance under this subsection and subsection (g)."
As for the interpretation of this statute, it means that essentially ANYONE can buy objects being liquidated from a federal collection. They call this process "Deaccession;" and usually the museum will see if another institution wants it first. But in practice, only federal employees of any degree are allowed to attend these events.. at least for the Smithsonian I was told by said friend. The Smithsonian hosts their own sale; many state-owned or other museums will yield the items to an auction house like Christy's or Sotheby's.
Further reading can be had here on the touchy subject of museum deaccession: