I did a google search and came up with this [1] which explains a lot. The push nodes only handle the actual sending of notifications, you need to setup the rest of the infrastructure yourself, as is explained in the link.
[1] https://github.com/Xzya/node-red-contrib-push/issues/2
I think if you want a local messaging solution then look at XMPP - plenty of Docker Containers (i.e. easy as to setup) with XMPP servers for your Local Lan - and then hook any XMPP clients into it that you want - there are node-red XMPP nodes available for sending and receiving messages
I am creating a real-time chat application based on node-red and much work have been done. The next step I thought to add is browser notifications and thoroughly searching on google I reached here. Can anyone tell if there is any new update or how can I implement this functionality. @craigcurtin@Colin@Trying_to_learn@janvda