How does a push actually work — does it keep my task in sync with the other app?

No. A push is a one-way hand-off. When you push an open task, the app copies it once into your chosen task app and then files it in this app's "Pushed tasks" folder. After that it's done: it's never pushed again, and the two copies are independent. Completing it here doesn't tick it off in Todoist/Reminders/Google Tasks etc., and ticking it off there doesn't change it here. Your task app becomes the place that task lives from then on.