It doesn't matter. House prices are widely known anyway.
The science of estimating house value is extremely sophisticated, given all the money that's at stake. Hide your price all you like; they can just look at comparables - the price of houses similar to yours in your neighborhood. They are looking at square footage, number of rooms/bathrooms, features (pool, boat launch etc.)
The only thing you can hide from comparable-comparers is hidden characteristics of your house that might significantly affect the price, like the shattered foundation that reduces the house to a scraper, or the hidden elevator in the garage that leads to the secret Bat-cave.
LLC's aren't as awesome as they seem
Common advice is "hide your purchase price, wrap the house in an LLC". Sure, OK, that has advantages, but it doesn't have this advantage. The sale price is still public record, just the LLC's name is there instead of yours.
If you really want to hide the sale price, find someone who has already created an LLC which owns their house... and purchase the LLC from them at a private price. In this case the house will not change hands; it will still be owned by the LLC. The hazard to this approach, however, is the system is tuned to protect homeowners from scams relating to transfer of houses. All sorts of swindles are possible in the sale of an LLC that are not possible in the sale of a house.
I can tell you, having sold a property owned by an LLC, the buyer had no clue nor interest about the LLC structure. My representative didn't really get it either, and didn't really pitch it well. So the buyer entered the deal expecting a straight sale, in other words buying the house from the LLC, not buying the LLC. Thus, the goal of averting the costs-of-sale was utterly defeated, even worse, all those costs fell on us as they traditionally fall on the seller. The buyer had no reason to want the LLC; his costs were exactly the same either way, and he went with the familiar. Maddening! So if you are a buyer wanting an LLC house, that may be a breath of fresh air to the seller.