There's more to this than just asking prices private versus trader.

Buying privately may not have the supposed protections a trader provides (assuming the trader is as honourable as the buyer imagines themselves to be) but the private buyer has the advantage of seeing the vehicle and the owner together and can make judgements about the owner during inspection and discussions about the vehicle and its maintenance, ie is the owner knowledgeable about the vehicle and its care or are they another white goods user who's run it continuously low on oil never serviced it and only washed in once in 3 years the week before the sale....look in their garage is it full of tools oils jacks etc? chances are they see some use.

Much depends on the type of vehicle being sold, something a bit unusual or specialist might be a better sale prospect for both parties private buyer from private seller especially if the vehcle has been subject to regular over and above TLC and both parties know what they are talking about and know what a vehicles true value is, which on older and unusual vehicles can range wildly between neglected tat and well cared for examples.

A 3 year old average mileage mass market car subject only to a typical manufacturers laughable (long life oil? give me a break) service regime, no, one would not expect to be paying as much for a private sale, buying from a trader you have little to go on as to the type of hands the vehicle has been in previously but with relatively new common cars maybe it matters less.