Have used Global Entry several times with nothing to declare, however, this trip we have bought a +$1000 rug, and not sure how to do this-by the paper from or on the computer screen?
Global Entry is separate from customs. On the screen, I think, there is a box for something to declare. Customs is after immigration and there should be a short line for Global Entry participants.
We declared something last summer--did it right on the computer screen. Then had to go into a separate line so they could check our bags. Still was quick and received the benefit of having GE.
When it asks you the series of yes or no questions on the Global Entry kiosk, one of the questions is, “Have you exceeded your $800 customs allowance for purchases abroad?” You then answer yes to the question and then your receipt will have an “X” and instruct you to see an an officer to sort things out. I actually did this for $950 in purchases once. After handing my receipt to the CBP officer and explaining to him what I had purchased, he decided not to assess me duty since I was only $150 over. He still said I did the right thing by answering yes to the question.
In some airports, immigration and customs are being combined. (AA at JFK). I thought this was strange as after you stepped out of immigration there was no one to check past baggage claim.
With Global Entry you don't need the paper form as the computer screen takes its place. As stated earlier, just note on the screen that you have something to declare and let them decide what to do. In many cases, if it's not a big deal and you are honest, they won't bother with the paperwork needed.
I have GE. I brought some tea and coffee into the U.S. last year and declared it on the screen. When the officer asked what food I was bringing in I told her. She sort of gave me the look as to why did I bother. I said it's technically food and I didn't want to break the law.