When sharing an airplane, what liability does the owner bear?
Most of the liabilities (but not all) fall under the insurance of the aircraft. Your insurance policy is your largest liability. This is important not just for protection of your aircraft, but if your insurance ends up void in an incident, you could have a lawsuit on your hands. Don't assume that aviation insurance is similar to car insurance, either.

Insurance


Pilot (on plan)- Aviation insurance has 3 catagories: Pilot, Named Individuals, Other (explained later). The owner pays insurance based on their qualifications and those specifically named. If you are going to be borrowing on a regular basis, weigh the cost of adding you to the plan.
Pilot (off plan) - Insurance specifies whether or not you may borrow the aircraft to others, but also what qualifications that pilot must have. You may have only have 100 hours and are insured, but the insurance company may ask for 500 hours of someone you borrow the aircraft to. 
Aircraft - Regardless of the pilot, if the aircraft is out of compliance on maintenance or airworthiness, it is not being flown legally and therefore voids the insurance. As the owner (primary on the insurance), this is your responsibility as an owner and the pilot's responsibility as PIC. Both of you can be hit for this one.
Reimbursement - Just because it is non-commercial use, your insurance may still specify what you can be compensated for. This may be only the actual expenses of the flight (fuel, oil, airport fees) and may include limits on hourly charges for maintenance. Keep in mind that this does not authorize you to break regulation on commercial use. You cannot make a profit.
Legal flight - Above all else, the flight must be legal. The flight needs to comply with all FAA regulations. 
Non-commercial use - I keep coming back to this, but this is very important. The pilot may be a commercial pilot, but this doesn't authorize them to use the aircraft for hire. With many insurance plans, you can have a CFI train you since you are the owner, but a CFI cannot train someone else and you cannot train someone else. This makes and aircraft for hire and requires a 100-hour inspection. 
Other - Pilots who do not own an aircraft may have their own (renters) policy. While this may cover liabilities of those in the aircraft, don't assume that this also covers your aircraft. A pilot paying $200/year won't likely have coverage for your shiny new Cirrus.


Other

As the owner, you have liability in accurate record keeping. If you fudge documents on your aircraft that make it appear FAA compliant to the PIC, then you can be held liable. 

Notes

These are only the items that come to mind. Feel free to add.