One of the reasons I don't tattoo my guys - figuring even most dog-savvy people (myself included) wouldn't flip a strange dog over if they found one running loose...With a golden tattoos are probably more visible than other dogs with dark skin...I guess that doesn't help with your question though - if I relied on a tattoo as my dog's primary form of identification I probably would keep the area clipped and easily (as possible) visible.
I had Sam tattooed for identification purposes. The tattoo stretched as he grew but it was still easily read when he was into his senior years. I used my SSN, which is a longer set of digits, so easily found when just rubbing his belly. Ike has a microchip.
Comet has a tattoo of his AKC number that the breeder had done. I registered it with AKC CAR, but it's not very visible unless you're looking. I plan to have him microchipped when he's neutered, just to play it safe. I've thought about clippering the area, but you'd have to really look for it anyway, so I figured a pound would take a careful look when trying to identify him and that clippering wouldn't make a difference either way.