Hi Zeb,
I think @oldturkey03 has already given you the information you need to figure this out; we just need to put it together in a way that makes sense to you.
Ultimately, the circuit breaker pops because there's too much current flowing through the circuit; more than it was designed to handle. Circuit breakers themselves do fail on occasion; it's rare, but it does happen, so it's not out of the realm of possibility that the breaker that's popping is old and weak and is letting go sooner than it should. So that's one idea.
Secondly, just because the problem microwave says it cooks at 1000W and you have another one that cooks at, say 1250W doesn't mean that's how much power it uses. According to the [link|https://www.manua.ls/magic-chef/hmm1110w/manual|Magic Chef HMM1110W User's Manual|new_window=true], the input power requirement is actually 1550W, much higher than the 1000W it cooks at. Looking at another example, there's a Toshiba 1250W microwave advertised on Amazon that only requires 1350W of input power. So the higher power microwave - 1250W - in this case actually takes ***less*** power to operate than your 1000W oven!
Let's do some conversions here. P = IE, the formula for calculating watts, Power (P) equals Current (I) times Voltage (E). Household voltage is 120V here in the US, and your circuit breakers are rated at 15 and 20 amps respectively. So 120V x 15A = 1800W, a bit higher (but not much) than the requirement for your microwave to run. 120V x 20A = 2400W, which in theory should be more than enough for your microwave, but apparently not, for some reason. Now that 2400W includes everything plugged into that circuit, so if you've got lights, refrigerator, etc., on that line, the maximum they can draw is 2400W - 1550W, or 850W without popping the breaker; maybe less if the breaker is tripping prematurely.
Ultimately, the only way to be sure is, as @oldturkey03 suggested, to actually measure how much power is being drawn. Get a power meter that you plug your appliance into and see what it's actually drawing. Use a night light and find out exactly which and how many outlets are being served by that breaker and test everything that's plugged in to that circuit. If it all adds up to less than 2400W, then your breaker may be faulty and should probably be replaced. On the other hand, you may find that the total power on that circuit is indeed exceeding 20 Amps when you toss the microwave into the mix.
Let us know what you find; you've got my curiosity as to what's going on with your microwave and I'd love to know how it turns out.