This was originally confirmed on a Q&A with IGN which can be found here. I thought this was pretty interesting since people who are aiming for the 1,000,000 gamerscore, cough cough, Stallion83, cough would continue to earn more and more gamerscore. If he didn't reach the 1 million mark by the time the Xbox One came out, he can continue going on until the 2 million mark. I'm sure this gets rid of all the worries if Stallion had any, and opens much more opportunities for him. Pretty good addition as well.
Via Eurogamer
"Every game has its own set of achievements, so if you're playing Call of Duty: Ghosts on Xbox 360 and on Xbox One, you can earn a separate set of Gamerscore in each game," he said.
Via Eurogamer