As an addition to vince comment: The tool in ArcGis has the reprojection as an extra step before the processing built in, you don´t see this(beside in the documentation). Qgis does not, and the standard tool does not warn you either (the grass tool for example tells you about it). When they are visibly onto each other it is just on-the-fly transformation. This is not as accurate and therefore not used for calculations. You have to live with that until Qgis changes the tool... Or you might put the to steps into the graphical modeler and get a tool that does it in one go without creating extra files.
And as sidenote: Watch out for the reprojection in general, there is more "science" to it than the easy use of the programs suggest. Depending on the needed accuracy and the eventually involved transformation (when the CRS are using different reference ellipsoids) the different programs might have different outputs in the range up to 1m.