giving apps permission to drive or gmail
If you give mailsapps or filemanagers on Android (those with a few millions users and likely trustable) access to your gmail-account or google drive, do they in general store these data (mails/files) on their servers as well? Or is it relatively safe to give them this permission?
Though this question is relatively incomplete and off-topic, I'll still try to answer it.
Whenever you login with Google on an application, you can find whatever access that specific app/service has over your Google account by visiting accounts.google.com and thus decide whether it's safe for your privacy or not.
That's it and one thing you should always keep in mind, doesn't matter it's a "free" service from Google or any other company,
"When something is free, you're the product"