...this user problem is a screen render issue, currently the nature of how indesign renders greyscale images in combination with drop shadows, even transparency blend space in cmyk, indesign doesn't take into account black ink profiles embedded...
...a comparison between photoshop and indesign visually will look different, if however you applied multi-channel to the grey image (add three white channels) and then to cmyk, a visual comparison will look the same as the cmyk profile is used, there is technically no greyscale in indesign. Important to have your adobe apps synchronized via Bridge...
...you can choose a cmyk profile as the grayscale working space inside of photoshop and use for both photoshop and indesign:
Color Settings > Working Spaces > Gray > Load Gray > find your profiles folder and choose the CMYK profile. Save to a color settings file for use in Bridge to synchronize to all Adobe apps.
Note about using the CMYK profile as the gray space for photoshop:
This method will give you a good preview match in indesign, assuming you use the same profile in InDesign as photoshop (calibrated monitor helps and is recommended to be done before anything else).
There might be a slight difference depending on cmyk profile you use.
CMYK profiles define black as something other than an absolute neutral. For example, US Sheetfed Coated has a slightly warm gray of LAB 10,0,1.
In photoshop, black grayscale is absolute black LAB 0,0,0, even with a CMYK profile like US Sheetfed Coated. In indesign, as there isn't a grayscale space, there's a slight difference at the shadows and midtones.
Andrew