I'd disagree and say he does. He knows things, only Cait knows, and is clearly able and shown to be able to listen in and see what it sees.
How did Reeve get all the information on DG and Omega, when he sent Cait Sith to Mako Reactor 0? Cait Sith never made it back. Furthermore, it got sucked into Nero's darkness. Unless Reeve was controlling Cait right then and there, and was looking through his eyes, there's no way he'd have gotten all that intel on DG's plans.
Furthermore, how did Reeve know Cait Sith in the Reeve puppet asked for Vincent's help and that Vincent declined? He carried on the plea for help as if he knew what Vincent's answer was before, when he spoke to Cait.
The creators explicitly say Reeve is controlling Cait and has likened him to a puppet master, so I don't see how you can really attribute Cait Sith as more than just an extension of Reeve, the real man.