Cats have classically been an ally of humans by getting rid of rodents like mice and rats. However, in this modern day and age, letting your cat hunt mice isn't necessarily the best idea. While it could potentially reduce the number of rodents you're dealing with, it could also harm your kitty. Here's why you should seek other ways to control your mouse or rat population.
Poison
If you're letting your cat hunt for mice, chances are you're not trying to simultaneously use poison to kill said mice. However, that doesn't mean that the rodents haven't been exposed to poison somewhere other than your home. Since most rodent poison doesn't take effect immediately, instead taking hours or days, a cat who kills, bites, or eats a rodent could potentially be exposed to the toxin. This could seriously harm or even kill your cat.
Diseases
Part of the reason why humans dread rodents is that they often carry diseases. Humans aren't the only ones susceptible to these diseases; cats can catch them, too. Rodents can carry nasty diseases like hantavirus, toxoplasmosis, or rat-bite fever. Worse still, some of these diseases can be spread from your cat to you, like toxoplasmosis. Cats excrete the parasites that cause toxoplasmosis into their feces, and when you clean it up, you could be at risk of catching the illness yourself, especially if you have a weakened immune system or you're pregnant.
Ticks and Fleas
Whether you keep your cats indoors or not, coming into contact with rodents can put them at risk of catching fleas or ticks. Even if you treat your cat regularly to avoid these pests, there's still a chance they might get them. Some medications only kill the pests, rather than preventing their eggs from hatching, or preventing further infestations. In addition, some anti-flea medications protect against fleas, but not ticks.
Ticks are especially a problem because they tend to use rodents as a stepping-stone to get to their final target. Once a tick is embedded in your cat's flesh, it won't let go until it either dies or you get help from a veterinarian (such as one from Canal Road Animal Hospital).
In short, it's not necessarily safe for cats to hunt rodents and rid you of infestations. While you may not be able to prevent cats entirely from killing an occasional mouse, if you know you have a rodent problem, try to tackle it yourself to reduce the health risk to your cat.