The European Organisation for Astronomical Research in the Southern Hemisphere (ESO) operates facilities in Chile because it offers good weather, good seeing, and it is in the Southern Hemisphere.

None of the ESO’s instruments are in Europe. They’re all in Chile. It would make no sense to place an instrument for Southern Hemisphere observation in Europe.

However, administration is out of Germany, and the captured data is processed by researchers mostly in Europe.

There are also many amateur remote hosting facilities for telescopes in Chile. It really has great atmospheric conditions. If my telescope is in one of these facilities, is it no longer my telescope?