The first question you should ask is what platform you want to send push notifications on: native, or web. Either way, you will need a dedicated back-end infrastructure to handle the traffic like the other answer says, but depending on the platform there are great third-party resources. For native push, I recommend using UrbanAirship. For web push, Roost provides a great service that covers Firefox, Chrome, and Safari. Building out the actual infrastructure required to send push notifications to all platforms is way more complex than it may seem, and using a third-party provider is definitely the way to go.

you need push server to handle push notification.

Pushwoosh, Ionic Push can handle this things when you don't need to setup server by yourself. or use https://github.com/Smile-SA/node-pushserver to setup push server to support APN, GCM.

So I will answer the question myself. There is no need to buy any other third party service. You can configure your own backend to send push notification with your choice of platform.