Unless You're Using Very Quick Kill Bait
Chances are the things will die somewhere else than right where you can see them. OTHO if you have decent sized infestation and or the odd ones were dying on their way to or back from a meal, then you'll see one or more.
That is the reason most exterminators, at least in NYC do not recommend using poison indoors unless it's totally necessary. More often than not rodents die in the walls, under floor boards, behind sinks and or large appliances (if not in them), and the resulting stink from decay can be horrible. One or two small mice might only create enough smell for a week or so, but if there are more or it's a rat (or god forbid several), the stench is just beyond words. Professional exterminators even have access to various chemicals that can be put inside walls to help deal with that problem.
The other thing which often happens is one starts seeing flies, even in dead winter. They are breeding/feeding on the decaying rodent's bodies and again it is horrible. Friends of ours had a bad mouse problem in their apartment building that resulted in finally the exterminator using an aggressive program of traps and poison baits. In the end the mouse problem was solved but every apartment and the common areas (halls, rubbish rooms, etc) were ful of flies. This was in December through the rest of the winter. No one could put two and two together until one of the other teanants who had lived through a similar situation before connected the dots. All that was left to do was hang lots of fly paper and wait for the plauge to end.
@supersurgilator,
Unless you've tagged, named or can postitively ID the mouse you chased out of your grandparent's garage, it probably is *NOT* the same one that is in your neighbors. While there may be the one odd mouse inside a home odds are there are usually more. In a given area, again it is rare there is just one mouse shifting house each time it's found out by various owners. It seems pretty likely the area is having a rodent problem and you'd be wise to contact local government and or ask around to see if anyone has reported similar experiences.
Most rodents by nature are burrowing animals, so anything that disturbs their underground homes tends to send them out into the open. Sewers, gas lines and other construction can turn what people thought was a rodent free area into something out of the plauge years.
Here in NYC certain types of construction that distrub earth must have the area baited before and during the project. Residents around the Second Avenue Subway line in Manhattan have reported an increase in both rats and mice since that project began.
It might be more proactive to set outdoor traps *around* your garage and see if that turns up anything. If you start cathing mice that way at least you know what you're dealing with.