I created a new debian stretch instance today (Nov 6), installed docker-engine, and ran:

adduser ebernhardson docker

Checked id after logging out and logging back in both immediately and after more than 30 minutes. In both cases it reports:

uid=3088(ebernhardson) gid=500(wikidev) groups=500(wikidev),1003(wmf),50062(project-bastion),50068(project-editor-engagement),50090(project-analytics),50120(project-deployment-prep),50174(project-integration),50276(project-search),50380(project-tools),52308(project-phabricator),52471(project-shiny-r),52857(project-discourse),53176(project-mwreview)

/etc/group on the other hand appears to have been adjusted appropriately:

docker:x:498:ebernhardson

For reference the appropriate line in nsswitch.conf defaulted to:

group:          files ldap

I had expected logging out and logging back in would put my user in this group, but it doesn't seem to have worked. I checked wikitech docs but was unable to find a solution there.

After