There was indeed a concentration/transit camp in Italy - the Risiera di San Sabba (which is located near Trieste, in the country's Northeast). However, this one was created and put in use only after Italy's fall in September 1943, when the area fell into German hands: it was the Germans who opened and run that camp... some 5000 among Jews and other 'undesirables (such as Communists, partisans, political dissidents) were interned and killed there.

While the Adriatic Operation Zone was run directly by the Germans, elsewhere in the Italian Social Republic... better yet, the areas that were under Fascist control... it was the G.N.R. militia who helped out rounding up (and in many cases, massacre) Jews, partisans, anti-Fascists and other civilians.

there was a very practical reason the camps were in east europe - that's where the jews were. and specifically poland, as that's where the vast majority of the extermination camps that people think about (all but two, handling the vast majority of killings). poland had the largest jewish population at the time, so it made sense to just round em up where they were.

there were many other types of concentration camps however all over europe including france germany etc, such as labor camps, pow camps, etc. these weren't the extermination camps that we all think about when someone says concentration camps, and jews weren't necessarily the only victims in these.

There is one in France called the Natzwiller-Struthof camp. Though it was built in 1941 and opened April of the same year in the Alsace region which had been annexed by the Reich before.

This Konzentrationslager is believed to have seen 52,000 people imprisoned and the death toll is about 22,000 or 25,000 people.

The idea came from Albert Speer who noted the presence of a rare kind of granit in Alsacian soils and decided that the place was a good one for placing a camp and make prisoners work here for the Deutsche Erd- und Steinwerke, Himmler's own company.

The Natzwiller-Struthof is in technically in France despite it having been built on annexed territory. This was by the way the first camp discovered by allied forces in the West and it is still present today as a museum.

Source : I've visited this place some years ago since it's not that far from where I live and we studied it. I don't think their own website would be a good source for here but if you enter the name in Google, it has a history section in several languages ( English, Italian and German ).