I have also double checked this and NO double dipping permitted. However I prefer to rack my mileage within Starwood program instaed of piling everything in CP+. Even if Starwood seems to give less, with all the bonuses offered you can rack a lot of points in a short period of time.
For example just before Christmas there was a promo for Platinum members (Stay 5 times and get addtional 15,000 points + the actual points for each stay etc...)
Using my two accounts (Starwood and CP) I can also get my vacation completely free by staying in a Starwood property and flying on a 1W member.... which is not bad