From my project root, where composer.json and composer.lock are, I tell composer to require drush/drush. It seems to do what it should, but upon completion it seems drush is not installed.
$ composer require drush/drush
Using version ^8.1 for drush/drush
./composer.json has been updated
> DrupalProject\composer\ScriptHandler::checkComposerVersion
Loading composer repositories with package information
Updating dependencies (including require-dev)
- Installing webmozart/path-util (2.3.0)
Downloading: 100%
- Installing symfony/var-dumper (v2.8.13)
Downloading: 100%
- Installing jakub-onderka/php-console-color (0.1)
Downloading: 100%
- Installing jakub-onderka/php-console-highlighter (v0.3.2)
Downloading: 100%
- Installing dnoegel/php-xdg-base-dir (0.1)
Downloading: 100%
- Installing nikic/php-parser (v2.1.1)
Downloading: 100%
- Installing psy/psysh (v0.7.2)
Downloading: 100%
- Installing pear/console_table (v1.3.0)
Downloading: 100%
- Installing consolidation/output-formatters (1.1.0)
Downloading: 100%
- Installing consolidation/annotated-command (1.4.1)
Downloading: 100%
- Installing drush/drush (8.1.5)
Downloading: 100%
Writing lock file
Generating autoload files
> DrupalProject\composer\ScriptHandler::createRequiredFiles
$ drush
The program 'drush' is currently not installed. You can install it by typing:
sudo apt install drush