Obviously, Zach, if both devices are a full load amps(FLA) you will trip the breaker if it is 15A. The miter saw will rarely hit the full 15A rated though. As you likely only run the miter saw a few seconds at a time, you may well get by with the situation.
That said, there would be less chance of tripping the breaker if both unit ran on separate circuits.
If both units ran at FLA for any period of time, and the breaker doesn't trip, then there is a likelyhood of overheating the wire and even possibly causing a fire.
I have found that hand tools are the best choice when I want to make mistakes at a slower rate of speed.
I don't suffer from insanity. I enjoy every minute of it.