You should also be able to build infrastructure, ports, air bases, AA, and forts in the territory of your puppets using your own IC. Not just radar. It's frustrating that you can't build air bases in Manchukuo as Japan, for instance.
I'm pretty sure this is a DLC-only feature. If you have Together For Victory, you can build in any subjects that are Colony status or better (I.E. not Dominions). With Death or Dishonor fascists have special subject types, and they can build in their subject's territory at any level. If you're playing the base-game only, then you can only build in territory you directly own.I've been able to do that in Singapore as Britain.
Mostly it works nicely, but radar is limited by the subject's technology, not the overlord's, so with an AI subject you'll often only be able to build level-one radar (and thus have terrible coverage).