Presumably like we do - (I'm heavily simplifying the complex regulation involved) the mitochondria have their own DNA which gets duplicated when the cell prepares to undergo mitosis, then the two new cells split the results.
There are lots of mitochondria in a cell (almost always, ignore red blood cells, for example), and the mitochondria have their own mechanisms for detecting when they need to divide (exercise and the resulting stress being one of them, which is why working out makes you more fit).
Also note that the DNA for most of the mitochondrial proteins is in the host cell's nucleus-- the mitochondrial DNA is stripped down to coding for just a few essential proteins.
I know the cytoplasm between cells is split on division and corresponding the mitochondria are roughly as well. They could technically have different times of reproduction than the overall cell and it would still work. I’m not certain what actually happens to the mitochondria during cell division though.