According to the Article VIII of the Outer Space Treaty:
A State Party to the Treaty on whose registry an object launched into
outer space is carried shall retain jurisdiction and control over such
object, and over any personnel thereof, while in outer space or on a
celestial body. Ownership of objects launched into outer space,
including objects landed or constructed on a celestial body, and of
their component parts, is not affected by their presence in outer
space or on a celestial body or by their return to the Earth. Such
objects or component parts found beyond the limits of the State Party
of the Treaty on whose registry they are carried shall be returned to
that State Party, which shall, upon request, furnish identifying data
prior to their return.
So the launching nation holds property rights and is responsible for space debris they generate. Five years later, the 1972 "Space Liability Convention" (in it's full title the Convention on International Liability for Damage Caused by Space Objects, PDF) was based on the Article VIII of the OST and further provisions it, but sadly isn't specific about space debris in any part. And establishing whose debris caused damage isn't exactly equal to establishing origins of some part that fell on someone's land and you can analyze in laboratories. It isn't even clear if orbital debris tracking facilities (say, NORAD) are legally obliged to share data on their origins in cases where it could hold the country running these facilities liable for damages to another country. And so on.
This and the fact that this issue is grossly under-regulated since mentioned first efforts were signed and ratified, and no internationally ratified law sufficiently addresses liability specifics as they pertain to orbital debris, comes with many interesting legal implications. But that's now over 47 years long and really tiring story so I won't bore you with all the details.
For a nice overview I'd suggest UNIDIR's Outer Space Security Conference Series, e.g. the Space Security 2014: Implementation and Compliance one has some really interesting talks on the subject. Also check for related activities under umbrella of the United Nations Office for Outer Space Affairs (UNOOSA).