A private sale can return more money, sometimes $1,000–$3,000 more than an instant offer, depending on the car. However, what it costs you is time (typically two to eight weeks), effort (photos, listings, enquiries, negotiations), out-of-pocket expenses (safety certificate, listing fees), and exposure to the risks outlined above.
A car-buying service like We Buy Cars pays a fair market rate, handles all the paperwork, requires no safety certificate, and pays you the same day. For most people selling a second-hand car, the trade-off is worth it. For sellers with time on their side and a desirable car in good condition, private can make sense. Only you can decide which side of that equation fits your situation.