On drupal.org, a distribution is a single download containing Drupal core, contributed modules, themes, and pre-defined configuration. You could create a distribution locally, using the same tools used on drupal.org to create a distribution, but that would involve creating a drupal-org.make file using
drush make, which is a command removed from Drush.
Instead, as Drush itself suggests, you should use Composer. Create a composer.json file, locally, and add all the dependencies (including modules) you need with
composer require. If then you need, for example, to create specific content types, roles, taxonomy terms, or other entities, you could create an installation profile, as described in How to Write a Drupal 8 Installation Profile. Using an installation profile doesn't work for already created sites, but for sites you still have to create, a custom installation profile helps in building a site.
As example of what an installation profile can do, see the content of the core/profiles/standard directory, in particular the config directory, which contains the configuration files necessary to create the default entities you normally used on the default Drupal installation.
An installation profile can have dependencies, a list of core modules that needs to be installed (see the standard.info.yml file), implement
hook_install() (see the standard.install file) and other hooks a module can implements, including the one to alter the site configuration form shown during the installation of Drupal (see the standard.profile file).