The devices are fairly small, so it's easy to isolate them from any conceivable unknown energy input. Electricity input can easily be monitored. Output can easily be monitored. If you have done a careful job of isolation, and the output over time is more than the same amount of mass could produce chemically (i.e., even a super-powered chemical battery), then you have a nuclear reaction. It's that simple.
No, input cannot easily be monitored, unless someone trusted to be impartial can provide and set up the lines.
You can have two different power lines going to the device, with each of them in addition to the two normal wires also has a third. The hidden wire is [+] in one wire and [-] on the other. The meters you clamp on to each of the cables will show a low current, not detecting the real power source.
And Rossi needs multiple low-power cables, and won't let others provide the equipment.
It isn't as though Rossi had one bolted to a table and wouldn't let anyone under the table to look.
Well, yes, that's exactly what it's like. The chain of custody is compromised, in that he does not allow the test to be set up in a 3rd party controlled environment, and doesn't allow them to replace any of the outside equipment that isn't part of the "trade secret" core.
If this was legit, Andrea Rossi could seal a unit, and send it to independent testing along with instructions for how to test it. It would be reproducible, and he would still keep his trade secrets. But he has refused this.