An InDesign document has no global resolution, so it's not "at 300 dpi" (or more accurately, pixels per inch [ppi]). Your InDesign elements are not twice as big as expected, they are 4.16 times the size. (Not) Coincidentally, that's the same factor between 72ppi and 300ppi. For InDesign-to-raster workflow (not entirely intended) one InDesign point = 1 Photoshop pixel at 72ppi.
Make sure the physical output dimensions of the ID and PS files match. So in Photoshop a 1600x3928 image at 300ppi outputs at 5.333"x13.093"—set the Document Setup for the file you are pasting from to the same dimensions. Or match the Photoshop output dimensions to the ID page's output dimensions. In ID a 1600 x 3928 pixel page outputs as 22.222" x 54.556" (1600 x 3928 at 72ppi in PS)
If you are on OSX you might find this script useful for replicatingID layers in Photoshop:
Awesome thanks Rob. That clears it up alot,