For various reasons our organization still uses ArcGIS 10.1. Our geodatabases have multiple feature datasets, but they all have a common coordinate system. Individual geodatabases represent specific sites across the country. I'm using the batch project tool to reproject several gdb that are in UTM WGS84 to the local State Plane NAD83. It doesn't seem to make a difference whether or not I apply a transformation or not during the reprojection, as the outputs behave the same in ArcMap. When adding to ArcMap, if I do not use a transformation in the data frame coordinate system, they don't align but will when I apply a transformation. It does not matter if the data frame is in WGS84 or NAD83. The transformation I'm using is WGS_1984_(ITRF00)_To_NAD_1983.
If I apply a transformation during the reprojection, shouldn't that negate the need to manually apply a transformation in ArcMap?