The filter function should sync across devices, however it's not available on the web right now. We will add it in future updates, though, but for now it's a mobile-only feature.
When will filters be available online? After the IOS updates I've found this really useful but a slowdown when using the web app. Todoist, your Karma is going down (J/K)
Other than that, keep up the great work!
Yes, we'll have filters available on the web-app soon, we're currently improving the UI to make it possible :)