How to copy current directory into Drupal profile folder via drush?
I actually had to specify extra type: "profile" (to follow this example), so type is specified twice (type of project, and type of download), as below:

api: 2
core: 7.x
projects:
  drupal7:
    type: "profile"
    download:
      type: "copy"
      url: "file://."


Then this would work fine:

$ drush make --no-core profile.yml drupal-7.39/
Beginning to build profile.yml.
drupal7 copied from file://..              


To avoid copying docroot/ it-self, it's good to actually create a separate folder for profile (as specifying wildcard won't work).