USDT and USDC have announced that in the future, their tokens will be used for payments only on their own networks, and that fees will be charged exclusively in their tokens.
With the way that the two stable coins have been well recognized, i don't expect that by now, they have not been assigned their own network, because they have gone wide to an extent of developing all it takes to have built their own network, lets take for instance, USDT, which has been available long time ago even before the introduction of USDC, by now it ought to have been on its own network than using other networks with the way they have been well recognized for use in cryptocurrency.