Using Pushwoosh for messaging app

Is it at all possible?

Yes, generally we have all the required instruments to do that including:

• Way to target individual users by their ID;
• Inbox to store the messages on the device

 However, there are certain limitations that should be taken into consideration.

1) You'll need to have your own sophisticated backend due to the following:

• sending messages directly from user's device is not secure;
• push technology is highly dependent on the good internet connection, so in order to rely entirely on push notifications for messages, you'll need a high-speed setup.

2) There is a limit to a payload size, which will shorten the allowed text length and the ability to send another content.

For example, in iOS a payload size is limited to 2K, for Android this is 4K. As you can see, the text will be pretty short, taking into account other technical data, included in the payload.

In order to overcome this obstacle, you may have the application to download actual content from your backend. 

3) Also note that while all the major players on the market (Whatsapp etc.) do use push notifications in their apps, however, they use it not for the chat itself but to notify their users.