I think push vs in-app are two different things, like email and telephone call. Push reaches people who are not using your app, but require OS-level permission. In-app gives you much more flexibility but they only reach people who are using your app (obviously!).

As to whether in-house or external SDK is best, it depends what you are trying to do and whether an external system handles it. For a typical retail environment the SDKs might be easiest--it's a well-established use-case. For something needing a more customized algorithm, an in-house system might work as well as an external SDK (for less money).
There are simple options to save you from starting completely from scratch, from PHP toolkits to Amazon SNS. It depends on what (if any) server environment you have...
Hope that helps!