I am considering studying a PHD in the states and I am wondering if i will have to pay international student tuition fees or will my research grant cover it (presuming I get the funding)?

Ask your grant awarding body - it all depends on their terms and conditions.

I'm currently working in a lab in the US and they are trying to persuade me to do a PhD here. They've told me that your fees are paid no matter which country you are from and that you get given the same amount of stipend as a US citizen.