The answer is that it depends on the flight.
A basic cross-country mission is a good lesson to share with a passenger who just wants to see a slice of your flight training. Make it a day that’s smooth and expected to stay that way.
The change of pace will benefit you too: Once aloft you will notice how the loading creates differences in aircraft performance and control responsiveness. For aircraft certified in both the normal and utility categories, the loading you calculate may put the aircraft out of the utility category, prohibiting certain maneuvers.
Just as well; any flight with plenty of ups and downs, like traffic-pattern work, or with higher load factors, such as a lesson on steep turns, or practicing emergency procedures, is something to avoid with an aeronautical neophyte on board.
Some passengers may insist that they are up for it. Experience taught me that it remains a 50/50 proposition, so be skeptical. You may not mind a bit of slipping and skidding in the (unlikely?) event that your rudder work to coordinate your other control inputs is slightly south of superb, but you are intent on watching your altitude, headings, and airspeeds; passengers have no frame of reference to take their mind off any discomfort.
Mild yawing sensations up front can be amplified in the back where the seating has a longer “arm” from the datum—and therefore a greater “moment”—than the pilots’ positions. (In weight-and-balance terminology, moment is defined as “the product of the weight of an item multiplied by its arm. Moments are expressed in pound-inches.”)
An exception to guidelines about when to avoid bringing someone along on a training flight arises if the individual is an observer—perhaps another student pilot or a CFI in training—rather than a casual passenger.
An observer might be better hardened against discomfort. But you still owe them your best efforts to provide a good ride and adhere to any manufacturer’s prohibitions on maneuvering based on aircraft loading or the center of gravity location.
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