You know when you've pretty much "gone local" by the fact that you don't have to think when using the local coins. Paper bills don't count as they are easily read and typically have the numbers on both sides of the bill (yeah, some people have trouble with bills as well, but they're just being slooooowwww :-)).
Coins are usually reused long enough to make them dirty and dented and thus hard to read and they typically only have the value on one side of the coin, so you have to look on both sides if you don't just know the value.
In addition, when you're using them you want to do so quickly because your in a line somewhere and there are people waiting behind you. So figuring out how many of each denomination fit into the others or add up to the total is not the time to look at each coin and spend time figuring it out.
Some people take the easy way out and just hold out a handful of coins hoping the cashier takes the right amount. Others just keep giving large enough bills so that they don't have to worry about counting, leaving the cashier to do all the heavy lifting.
I, on the other hand, am here in Japan on my last stop on my trip-around-the-world and I find myself just working in the local currency, including things like adding an extra 10 ¥ to a ¥1,000 bill when asked for a ¥160 fare for the metro. I'm proud of myself :-).