A single source of randomness drives the price, so once the price and time are known, the volatility is determined. If the volatility is constant, the model is the same as the Black-Scholes model.
Consider a hypothetical example in which the local volatility is 0.20 × √(100 ÷ price). It is 20% at a price of 100 and about 22.36% at 80. Using a function fitted to the market smile, the same model can reproduce the prices of options at all strikes and expiries. On the other hand, how the smile moves when the underlying price moves is determined by the model, and it moves in the direction opposite to the market.