My incredibly scientific timing (one one-thousand, two one-thousand ...) tells me that opening the sidebar is approximate one second to open and everything else on the sidebar basically operates not instantly, but near enough that there's no way for me to experience any kind of a practical delay.
What I mean by that is, I can navigate the sidebar without waiting for anything to load. I press a direction or button, and the response is close enough instant, that I don't have to wait to enter my next input.
Edit: I also don't have an amazing Internet connection. It's a pedestrian 30/5 connection, and using the Xbox Live network test it shows I have about 60ms of latency to Live. No packet loss. So not huge bandwidth and not amazing latency.