So, the answer of this questions heavily depends on the when, how, and who, as so many things in history. Also, it needs some contextualization in terms of what happened to the personal possessions before they were liberated by the Allies along with the Concentration Camps. This is necessary because what the Allies found there was only a very small portion of the overall total of personal property confiscated by the Nazis from their victims.

The Holocaust and especially the camps were a for-profit venture by the government of the Third Reich. As per German law especially created for this, Jews, which were classified not as citizens of Germany but rather as "Schutzbefeholene" (akin to wards) would lose all their possessions once they left the border of the German Reich and they would in turn revert to property of the German state. The same applied to other territories, only with the property going to states such as Vichy France, the Nazi occupation of the Netherlands and so forth. The personal possessions people brought to the camp were in the majority of cases collected in the camp magazine (Kanada in the internal speak of the inmates) and regularly packaged and brought to Berlin where the moveable possessions such as shoes, suitcases etc. were either auctioned off or distributed to the Winterhilfswerk (a German charity organization to help those Germans in need, collect materials for the Wehrmacht and distribute aid to bombed out Germans). There were two special cases: Gold rings, jewellery etc., and gold teeth broken from the mouth of dead victims were transferred to the Degussa company to be smelted in gold bars, which in turn were brought to banks in Switzerland and Germany, and the cut hair of victims, which was bought by a German company producing felt articles.

What the various Allied forces found at the camps they liberated was what the Nazis didn't have time to ship and distribute to Berlin in order to be made into money. However, the Americans also found at the Kaiseroda Mine at Merkers vast quantities of gold bars, jewellery and other valuables shipped there by the Reichsbank in the last weeks of the war to hide the assets from the Soviets approaching Berlin. This "Nazi gold" and a variety of other valuables (paintings, jewellery, foreign currency and so forth) were transferred together with valuables liberated by the Western Allies in the camps given to the so-called "Safehaven Program", a program initiated by the Departments of State and the Treasury and the Foreign Economic Administration in 1944. This was created with the aim of identifying and stopping the movement of Nazi assets out of Germany and then to locate, recover, and restitution them.

This program, which was also concerned with the gold the Nazis had taken from the various central banks of occupied countries, then transformed into the so-called Tripartite Commission for the Restitution of Monetary Gold, a commission composed of American, British, and French representatives and concerned with the restitution of primarily gold and currency to both private individuals and state agencies of formerly occupied countries. This was especially pertinent for the London Agreement of the early 50s that regulated a lot of restitution issues vis-a-vis states and private individuals of persecuted groups in lieu of an official peace treaty.

The non-gold, non-currency moveable assets of victims of the camps however remained mostly in possession of the various memorial sites set-up after the war in former concentration camps. To this day, people can and do claim ownership of certain items (such as suitcases of relatives) and they are reconstituted but all in all, they are mostly kept for preservation purposes.

This is also true of the camps liberated by the Soviets that now hold possession of moveable objects of victims (e.g. the shoes on display in the USHMM in Washington are on loan from the Majdanek memorial site) but are willing and do restitute certain objects to relatives of deceased persons to this day.

The issue of gold and jewellery in the Soviet liberated camps is a bit of a tricky matter however: Originally slated to also be restituted to individuals or relatives they were robbed from, like in the West, the Cold War and emerging Soviet and socialist policy towards these countries bearing no responsibility for the crimes of fascist predecessor states took hold and oftentimes that lead to these items being completely forgotten. E.g. it was only last year that a massive amount of items from Auschwitz in the possession of the Polish academy of science was rediscovered.

There are restitution battles still to be fought (especially concerning art and also some stuff still in Swiss bank accounts) but for that there would need to be someone claiming these items and given the comprehensive catastrophe of the Holocaust, many families were just completely wiped out with anybody who could claim ownership dead. So, many of these items remain in possession of memorials and other state institutions, most likely to never be reclaimed by anyone.

Sources:

• U.S. and Allied Efforts To Recover and Restore Gold and Other Assets Stolen or Hidden by Germany During World War II, 2007.

• Martin Lorenz-Meyer: Safehaven: The Allied Pursuit of Nazi Assets Abroad.

• Dan Stone: The Liberation of the Camps: The End of the Holocaust and its Aftermath