I am using QGIS 2.10.1 with some Chicago data (from https://data.cityofchicago.org/browse?tags=gis) and the Google Streets plugin.
My issue is with the CRS's for the Chicago layers and the Google Streets layer. Google Streets is in WGS 84/Pseudo Mercator (EPSG 3857) and the other layers are in a User Defined Coordinate Systems CRS. When I leave them as is, they line up perfectly on top of each other, everything matching where it should be. However, if I change the Chicago data CRS to WGS 84 it moves the vectors to Central Africa respective to the Google Streets layer.
What is the reason for this? And can I still perform meaningful analyses using both the Chicago data and the Google Street layer together with different CRS's?