It is written here that the push notifications will work even if the browser is closed, but I tested it and it is not the case. I receive push-notifications only if the brow
On desktop the browser needs to be running since that is the process that receives the push messages. Some extensions, like hangouts force the browser to keep running even when the last tab is closed so for users with one such extension installed push will work all the time.
On Android, the browser does not need to be running since the entity in charge of receiving the messages is baked into Google Play Services.
The Chrome team is working on the desktop issue in https://bugs.chromium.org/p/chromium/issues/detail?id=402456