Private stock prices are the last price someone was willing to buy that stock at.

It is slightly less simple, because you may give a discount for advice, but private stock prices are definitely reliant upon supply and demand. If one investor wants to set your valuation at $1B and another at $100m, you go with the one who offered a $1B valuation. There is occasionally a discount for advice/connections, but the mechanism that the highest bidder wins is generally the same. Private market stocks are by no means exempt from economics or supply and demand.