Assumption
My question assumes the following situation, that is what I could understand about Google's push notifications system. If I misunderstood anything, please correct me here.
When a web service wants to send a notification to a user who installed its app (client), it sends the notification content to Google's push notification server. The server then contacts the user's device instructing it to show the notification.
This system prevents user's device from continously checking for notification, that would cause battery drainage and other annoying side-effects. However, some privacy concerned people point out that it allows Google to access all the notification contents, including for example private messages.
Question
If I have an app (client) that manages* notifications on my device for a web service and doesn't have an option to disable them, the only way I can prevent them to be shown is through Android settings.
To my understanding, if I do so the web service (that doesn't know my Android settings) will still send notification contents to Google's server, thus allowing Google to know notification content, but the server won't send it to my device. So, disabling notifications through Android settings wouldn't have any effect on privacy. Is that right?
*I use the word "manage" because usually the notifications are shown if and only if the app is installed and it often has settings for them (sound selection, vibrate and so on); however, I don't know how much notifications depend on the app and how much on the system. Anyway, it looks to me that it is not very important for the question (but I might be wrong of course).