The Spiritiualist's Phantom resides in the Spiritualist's mind while it isn't manifested, and it has its own mental stats separate from yours, so could the phantom attempt its own mental skill checks separately from you?
An unmanifested phantom probably can't make skill checks
Any effects beyond those described by the special ability shared consciousness would have to be adjudicated by the GM. As rules already exist that describe the benefits of an unmanifested phantom, persuading the GM that there should be even more benefits—like the spiritualist being able to make a bunch of duplicate skill checks because there's a ghost in his head—may be difficult. Good luck.