I am guessing that out of Adobe Illustrator, Photoshop, and InDesign, you would use Photoshop?
Yes, this is best done in Photoshop. This is known as effects processing of imagery in that:
(on the left) a desaturation effect has been applied.
(on the right) an exaggerated halftone effect.
Masking is required to isolate the effects to portions of the imagery, something that Photoshop is good at, even if the UX is clunky, retarded and antiquated.
Additional commentary on some of the steps to achieve the desired effects upon an image in Photoshop
You can accomplish this layering effect by duplicating layers in Photoshop and applying a Layer Mask to pick out the parts of the photo you want to apply the monochromatic coloring to or the halftone effect. To do a monochromatic coloring other than black and white you go to Image Adjustments and should be able to alter the saturation of the photo on the left side and "colorize" it to make it monochromatic. Hopefully this helps!