You put a spectrum alalyzer inside and transmit to it.

This guy shows that a box of aluminum foil drops EM by at least 40 dBm. Chicken wire is more like 20.

https://youtu.be/9M6h1U9jgWs

A simple test would be to stick a few wireless devices in the box and check that they all loose signal once the box is closed. If you want to get fancy any place that does EM interference tests of consumer goods should have the room and equipment to test the box across a broad spectrum.

My intuition would be that the lid doesn't seal that good as it is. For long-term storage that's easily fixed by putting the shielding on the outside and gluing it shut with copper tape. Maybe you could instead add some copper-lined magnetic flaps that close the gaps.

I think various electronics companies have test rooms for this and govements have actual EPM test rigs

This is not too healthy for the microwave