There's an article today in the New York Times, "Museums Under Fire on Ancient Artifacts". (the NYT requires registration to view the article). It touches upon some interesting points about artifacts and their acquisition. The issue of restitution has been going on for many years now, but recently there has been an increasing call for the return of various ancient artifacts by certain countries (Greece, China, Italy...) Some of these items have been donated to museums by private collectors and some have been purchased acquisitions.
The questions is, how were they obtained in the first place? I did some research on this matter a couple of months ago after reading another article and it's pretty clear that most museums around the world don't have a strict set of guidelines as to how they verify the source of an artifact. Some have recently adopted more stringent rules regarding the issue, but others haven't yet.
In 1970 a Unesco convention prohibited the "circulation of illicit antiquities" Some of these artifacts have been acquired decades ago, way before 1970. The art black market is extremely vast and as far as I can tell, in the past, if the piece "someone" was offering either a private collector or a museum curator was an especially significant one, then the source might not have been questioned. The museums that have adopted regulations have typically chosen a "from" date (any artifacts acquired prior to that date are not subject to the regulations). The problem is, how do you decide what date to set as the "cutoff" date? The date of the Unesco convention? Or, the date your country joined the Unesco convention (in case it was later)? It's all so fuzzy because no international law is in place for something like this. And even if there was, what about enforcement?

One of the Elgin Marbles:
Horsemen from the west frieze of the Parthenon
(Image from British Museum website)
On one hand, there are countries asking for their national treasures to be returned. It's getting serious too, museums are being taken to court to answer for artifacts that are added to their collections. The former curator of the Getty Museum is facing trial in Rome charged with "conspiring to traffic in looted art". Greece demands the return of the Elgin Marbles (sculptures removed from the Parthenon and taken to England in the early 1800's, and are currently housed at the British Museum) On the other hand, we have the museums arguing they provide a means for these artifacts to be appreciated by a much wider audience by having the artifacts distributed around the world rather than in one country.
[Aside: yeah, but let me tell you "distributed around the world" is a generous term for what the actual distribution probably looks like]
It brings up all kinds of questions. Are ancient civilizations, world heritage? Or should the great ancient cultures be regarded as "belonging" to the modern countries which hold the political boundaries? Even if one thinks the artifacts should be returned there is a problem because the precise demarcations and boundaries of a modern state would not necessarily match up in antiquity. The territory belonging to Greece in ancient times for example, comprises various modern day countries. The same goes for pre-columbian cultures like the Aztecs, Mayas, Incas, etc. and others around the world. So if there were to be a mass restitution of artifacts from all the museums of the world, how do you decide what goes to which country? I know that's not the central issue, but still.
There are also other factors to consider. These artifacts need proper conservation. Some museums have the financial resources in order to take better care of them. That is not to say that more modestly equipped museums in less affluent countries should be deprived of having and displaying such treasures. On the downside, museums with a lot of funding can offer to pay millions of dollars sometimes for very valuable pieces, which is an incentive for the illegal trade.
It ceases to be a matter of preserving the world cultural heritage when museums are in cut-throat competition with each other, offering enormous sums of money for artifacts. In the same way, it's no longer a question of "bringing home" what is yours when you would have it sitting in a basement gathering dust and deteriorating as long as they're "home".