Can I bring a person with me while getting landing current if they're a licensed, current pilot?
As long as  you are not the legal pilot in command you can do what you want.  This means the other pilot has to be willing to act as the legal pilot in command on this flight.


  61.57(a) General experience.
  
  (1) Except as provided in paragraph (e) of this section, no person may
  act as a pilot in command of an aircraft carrying passengers or of an
  aircraft certificated for more than one pilot flight crewmember unless
  that person has made at least three takeoffs and three landings within
  the preceding 90 days, and—
  
  (i) The person acted as the sole manipulator of the flight controls;


The regulation is clearly required of the legal pilot in command.  If your current passenger is willing to act and take responsibility for your flying as the legal pilot in command then it is allowable.  

Unfortunately, the other pilot is taking all the responsibility and gets none of the credit as they, under an airplane certificated for single pilot operations, cannot log any PIC time while you are getting current.

From LOGGING PILOT IN COMMAND TIME (FAA):


  However, two pilots may not simultaneously log PIC when one pilot is
  sole manipulator of the controls and the other is acting as
  pilot-in-command if the regulations governing the flight do not
  require more than one pilot.




If the other pilot is a CFI and it is a training flight then no one has to be landing current as there are no passengers.  See the Kortokrax interpretation.

Relevant excerpt:


  We agree that, for purposes of section 61.57(b), an authorized
  instructor providing instruction in an aircraft is not considered a
  passenger with respect to the person receiving instruction, even where
  the person receiving the instruction is acting as PIC. (The instructor
  must be current, qualified to instruct, and hold a category, class and
  type rating in the aircraft, if a class and type rating is required.)
  The instructor is not a passenger because he is present specifically
  to train the person receiving instruction. Neither is the person
  receiving instruction a passenger with respect to the instructor. This
  training may take place, even though neither pilot has met the
  61.57(b) requirements.