If you go to a Lab prepare yourself with 3K+ Euro minimum. And if some of the tests fail prepare another 3K...
It is not required by law that you go to a Lab. You can do self certification and issue the certificate signed by you.
You need to prepare a so called "Technical File" . This technical file is a compilation of your schematics, PCB design and mechanical drawings if any. Also you must describe the design rules that you apply to make sure the design follow the good practices of EMC.
Then you must buy a cheap Spectrum Analyzer (Rigol or similar) and a couple of Near Field probes that you can buy online for 150 to 200 Eur. Connect the probes to the spectrum analyzer and observe the radiated emissions. Compare with the standard's (EN directives) limits. You have also to test the conductive emissions, if your device is connected to the mains.
Next you must test RF immunity: Buy a RF broadband generator with at least +20dbm power, and connect the Near Field probes to it. Move the probes around you circuit and see if it fails.
Another way is use several radio emitter's a CB Radio (27MHZ), amateur Radio (144MHZ) and hit the transmit button near your device. This transmitters have several watts of power and can be used to test immunity to several frequencies (harmonics also help here). Use your mobile for the higher frequencies.
Go to your local supermarket and buy a lighter (the one that are used to light gas ovens). This devices emit a very large electric spike that you can use to test static electricity immunity.
AFAIK, no one can request your technical files except the authorities. And normally they don't bother you if you device does not produces interference. But if you bother the competition they can report you just to make you prove your designs, but I never seen this happen on my country.
Google for "CE pre-compliance". You will find lots of interesting things.
Best of luck