Role of prior scales when not using Stan?

In a Bayesian paradigm, it is possible to use prior distributions and explicitly define their parameters. And we use Stan's MCMC sampling methodology for finding the posterior distributions of parameters given data. This is clear to me. But it is less clear how Newton method make use of the prior scale parameters that are given as parameters. How do they play a role in the optimization when the fitting falls back to Newton-based methods, or when algorithm is directly set to Newton? Thx.