Hi.

What does it take to call yourself a professional LEGO builder? If you are paid to design and build MOC's, does that count?

You'd need permission from LEGO to use their name to describe your business.

You have to attend LEGO's official courses and be certified to officially advertise yourself as an authorized LEGO professional. The process is pretty rigorous and last I heard they only accept a handful applications every year, anyway. Otherwise of course you can offer your services, you just cannot use their trademarks and certain words. You could write "Years of experience with brick-based models on a professional level", but not much else. It sucks having to dance around words, but LEGO are known to send their lawyers even over tiny misrepresentations of their brand and with an exclusive club like the certified builders they'd shoot you down quick and hard.

Mylenium