"The driving force behind a private service's price is what they believe the maximum amount you will be willing to pay for their service."
Well, no. Not exactly. The driving force is supply and demand. That's why you see gas prices skyrocket during natural disasters -- it's the market's way of compensating for the fact that there's less of it available. If you want gas, you can get it, but only if you're willing to pay higher prices. In that way, you keep people from hoarding precious goods when they don't need to.
That's also why when you see government enact price controls to 'protect' the consumers, the words 'shortage' inevitably appears shortly thereafter. The price controls keep the prices artificially low, so more people buy. So much more that the 'supply' of that good or service isn't enough to handle the increased demand, and there's a shortage. That shortage otherwise wouldn't be there (or would be very, very minimal) if the market were allowed to set prices according to the situation.
That's also the inherent problem with government: They're a monopoly, and they can't respond to changing situations like the private sector can. That's why you have problems of rationing during droughts. The government says to ration because that's the only way to handle the fact that there's a decreased supply of water but the same demand. If the price were to go up, people would naturally curb their desire to water their lawns as often as they otherwise would, all without the threat of imprisonment or force.
The other problem with your statement is, "How do they know how much it costs?" There's no inherent 'right price' for a gallon of water. It all depends on a collection of factors -- factors that no one person can possibly know (see also: Why central economic planning fails); and that collection of factors is what you take into account when you have multiple producers competing to provide a product or service.
