It's very nearly true, but for clarification:
You need a general license (which in effect everybody already has). However, you do have to register the trap with the police and get a number for it. You then put a notice on it that says something like "this is a legal trap, license number #12345. If you suspect it is being used illegally, please contact....." (the text to use is on the BASC website). You also have certain obligations to adhere to, such as the size of trap, and must ensure the call bird has food and water at all times, and is checked I think twice daily.
The police can come and inspect, but in reality they have bigger fish to fry unless they receive a complaint. (The best way to avoid this of course is to keep the trap out of view of passers by, and to make sure you absolutely obey all the rules).
Now, having said all that, I haven't used our trap for a couple of years now (it was only ever a last resort). We had a major problem with magpies when we moved in here, but having reduced the excess of population, everything seems ok just now, and there are definitely more of the wee birds about.