Will there be a privacy partition in the Crew Dragon during the one week trip around the moon?
If the Dragon flies without a service module, which I believe is the current plan as no service module has yet to be unveiled by SpaceX, then one should not expect any kind of privacy during the journey.
For example, the Soyuz toilet is simply an apparatus which is removed from storage when needed, cf.. Even in the case of the ISS, which has orders of magnitude more volume than the Soyuz, astronauts have a "closet" as a bedroom and the toilet is also barely private: .
One of the benefits of the Soyuz design is that there are two modules, Orbital module and Descent modules that are habitable and thus, there is 'another room' to get some privacy.
Dragon does not have this option. Nor did Apollo.
For short term missions, people can deal with it. For longer term some kind of add on modules is usually suggested. A small Bigelow habitat, a hard skinned module, or somesuch that provides extra space, and a second room for some privacy is option.