Yes, in certain circumstances. FAR 61.113(a) tells us that “no person who holds a private pilot certificate may act as pilot in command of an aircraft that is carrying passengers or property for compensation or hire; nor may that person, for compensation or hire, act as pilot in command of an aircraft.” Fortunately, that isn’t the end of the story. In addition to other exceptions, Section 61.113(c) says “[a] private pilot may not pay less than the pro rata share of the operating expenses of a flight with passengers, provided the expenses involve only fuel, oil, airport expenditures, or rental fees.”
Operating expenses fuel and oil consumed on a flight and ramp or tie-down fees at the destination airport, or if the aircraft is rented from an FBO or other third-party, the hourly rental fee plus the cost of fuel, if that is not included in the rental fee. These expenses may be shared. However indirect expenses such as insurance, maintenance, depreciation or other capital costs (e.g. costs of ownership) can not be shared.
Although the amount the pilot and passenger(s) must each pay doesn’t need to be calculated penny for penny, according to the FAA if the private pilot pays less than an equal share of the total operating costs or flight, the FAA will assume the private pilot was not actually sharing expenses. So, if you divide the total allowed costs for the flight by the number of people in the aircraft during the flight that should get you close enough to satisfy the FAA.
Also, don’t forget that the private pilot and passengers must have a common purpose for the flight as well: they must be flying to the same destination for a common reason. If the private pilot doesn’t have any business at the destination, that wouldn’t be a common purpose flight. Similarly, building flight time or getting flight instruction isn’t a common purpose unless that is the purpose of the passenger(s) as well.
So, as a private pilot you have the opportunity to share expenses in certain limited situations. But if you don’t meet the requirements of those exceptions and the FAA finds out, you could be looking at a certificate action.