I'm no lawyer, so I can't be definitive of the legalities, but my understanding is that you only need a written agreement with the publishers if you plan to sell the transcription. GuitarTricks has such agreements with all the publishers and songwriters whose songs are on the site.
Otherwise, you are free to transcribe (and play) any song you like however you choose!
Mike
Keep rockin!
Mike Olekshy
GT Guitar Coach